## The annular-sum and Hale-McClellan methods of 3D wavefield extrapolation

### Gary F. Margrave, Saleh M. Al-Saleh, Michael P. Lamoureux, Wentao Sun

A new theory of efficient 3D wavefield extrapolation, patterned after the established Hale-McClellan theory, is presented. The new method uses annular-sum filters, operators that compute the sum over a specific annulus of all wavefield samples at each point, weighted by the radial samples of the 3D wavefield extrapolation operator. In contrast, the Hale-McClellan algorithm uses samples of the 2D wavefield extrapolation operator to weight the wavefield as filtered by McClellan filters. The latter are shown to be a type of annular-sum filter that incorporates an additional approximate 45 degree phase rotation and an amplitude decay. The annular-sum filters can be computed exactly in the 2D Fourier domain by multiplication by a Bessel function. They can also be computed approximately, but accurately, by spatial convolution with a small spatial operator. The annular-sum filters are used to formulate the Wiener least-squares design problem for 2D circularly symmetric filters as a standard matrix-vector problem that can be solved by standard solvers. The elements of the radial convolution matrix are shown to be the result of annular-sum filters applied to the 2D impulse response of one of the circularly symmetric filters.