The waveform in equation (4) often arises in practice (as the 2-D Huygen's wavelet). Because of the discontinuities on the left side of equation (4), it is not easy to visualize. Thinking again of the time derivative as a convolution with the doublet ,we imagine the 2-D Huygen's wavelet as a positive impulse followed by negative signal decaying as -t-3/2. This decaying signal is sometimes called the ``Hankel tail.'' In the frequency domain has a 90 degree phase angle and has a 45 degree phase angle.
In practice, it is easiest to represent and to apply the 2-D Huygen's wavelet in the frequency domain. Subroutine halfdif() is provided for that purpose. Instead of using which has a discontinuity at the Nyquist frequency and a noncausal time function, I use the square root of a causal representation of a finite difference, i.e. ,(see any reference on Z-transforms) which is well behaved at the Nyquist frequency and has the advantage that the modeling operator is causal (vanishes when t<t0). (Fourier transform is done using subroutine ftu() from PVI.) Passing an impulse function into subroutine halfdif() gives the response seen in Figure 1.
# Half order causal derivative. # subroutine halfdif( conj, n, x, y) integer n2, i, conj, n real omega, x(n), y(n) complex cz, cv(4096) n2=1; while(n2<n) n2=2*n2; if( n2 > 4096) call erexit('halfdif memory') do i= 1, n2 { cv(i) = 0.} do i= 1, n if( conj == 0) { cv(i) = x(i)} else { cv(i) = y(i)} call ftu( +1., n2, cv) do i= 1, n2 { omega = (i-1.) * 2.*3.14159265 / n2 cz = csqrt( 1. - cexp( cmplx( 0., omega))) if( conj != 0) cz = conjg( cz) cv(i) = cv(i) * cz } call ftu( -1., n2, cv) do i= 1, n if( conj == 0) { y(i) = cv(i)} else { x(i) = cv(i)} return; end
hankel
Figure 1 Impulse response (delayed) of finite difference operator of half order. Twice applying this filter is equivalent to once applying (1,-1). |