Nonstationary filters, pseudodifferential operators, and their inverses

Gary F. Margrave, Robert James Ferguson

An inversion scheme for nonstationary filters is presented and explored. Nonstationary filters can be inverted provided that the nonstationary transfer function is known and invertable. The two fundamental nonstationary filter types, convolutional and combinational, play a complementary role in the inversion process in that the inverse of nonstationary convolution is nonstationary combination with the inverted transfer function, and vice-versa. These concepts lead to simple expressions for forward and inverse Q filters, forward and inverse wavefield extrapolators, and for any other circumstance where the nonstationary filter form is known.

Inverse Q filtering with nonstationary combination is very precise and provides a simple analytic formalism for such filters. When done with nonstationary convolution, the result is much less acceptable. This assumes the forward Q filter was convolutional.

Numerical experiments with two different wavefield extrapolators, NSPS and PSPI, are shown for a variety of velocity models. NSPS is a convolutional nonstationary extrapolator while PSPI uses the combination form. For constant or weak velocity gradients either extrapolator can invert the other. When velocity becomes chaotic (random), NSPS is required to invert PSPI and vice-versa. Inversion results also improve, even for chaotic velocities, when the extrapolation step size is decreased. A series of small extrapolation steps can be inverted much more successfully with another series of small inverse steps than with a single step. As step size decreases, so does the distinction between NSPS and PSPI and either one becomes able to invert the other.