Waves and Fourier sums

THE HALF-ORDER DERIVATIVE WAVEFORM

Causal integration is represented in the time domain by convolution with a step function. In the frequency domain this amounts to multiplication by $1/(-i\omega)$. (There is also delta function behavior at $\omega=0$ which may be ignored in practice and since at $\omega=0$, wave theory reduces to potential theory). Integrating twice amounts to convolution by a ramp function, $t\, {\rm step}(t)$, which in the Fourier domain is multiplication by $1/(-i\omega)^2$. Integrating a third time is convolution with $t^2\, {\rm step}(t)$ which in the Fourier domain is multiplication by $1/(-i\omega)^3$. In general

$$t^{n-1}\ {\rm step}(t) \quad =\quad {\rm FT}\ \left( { 1 \over (-i\omega)^n} \right)$$ (28)

Proof of the validity of equation (28) for integer values of $n$ is by repeated indefinite integration which also indicates the need of an $n!$ scaling factor. Proof of the validity of equation (28) for fractional values of $n$ would take us far afield mathematically. Fractional values of $n$, however, are exactly what we need to interpret Huygen's secondary wave sources in 2-D. The factorial function of $n$ in the scaling factor becomes a gamma function. The poles suggest that a more thorough mathematical study of convergence is warranted, but this is not the place for it.

The principal artifact of the hyperbola-sum method of 2-D migration is the waveform represented by equation (28) when $n=1/2$. For $n=1/2$, ignoring the scale factor, equation (28) becomes

$${1\over \sqrt{t}} \ {\rm step}(t) \quad =\quad {\rm FT}\ \left( { 1 \over \sqrt{-i\omega}} \right)$$ (29)

A waveform that should come out to be an impulse actually comes out to be equation (29) because Kirchhoff migration needs a little more than summing or spreading on a hyperbola. To compensate for the erroneous filter response of equation (29) we need its inverse filter. We need $\sqrt{-i\omega}$. To see what $\sqrt{-i\omega}$ is in the time domain, we first recall that

$${d \ \over dt} \quad =\quad {\rm FT}\ \left( -i\omega \right)$$ (30)

A product in the frequency domain corresponds to a convolution in the time domain. A time derivative is like convolution with a doublet $(1,-1)/\Delta t$. Thus, from equation (29) and equation (30) we obtain

$${d \ \over dt} \ {1\over \sqrt{t}} \ {\rm step}(t) \quad =\quad {\rm FT}\ \left( \sqrt{-i\omega} \,\right)$$ (31)

Thus, we will see the way to overcome the principal artifact of hyperbola summation is to apply the filter of equation (31). In chapter  we will learn more exact methods of migration. There we will observe that an impulse in the earth creates not a hyperbola with an impulsive waveform but in two dimensions, a hyperbola with the waveform of equation (31), and in three dimensions, a hyperbola of revolution (umbrella?) carrying a time-derivative waveform.

Waves and Fourier sums