THE IMPULSE RESPONSE OF HALE'S DMO OPERATOR

The purpose of this appendix is to explain the apparent difference between Hale's DMO operator Hale (1984) and operator (17), which is the first term in the offset continuation to zero offset. The difference between the two operators is simply the real part of Hale's DMO. Therefore, I will analyze the real and the imaginary part separately and discuss the contributions of each. In order to do this, I derive the impulse response of Hale's DMO in the time-and-space domain, following the results of Stovas and Fomel 1993. The high-frequency asymptotics of the impulse response has been investigated previously in a number of publications Berg (1985); Hale (1991a); Liner (1990). Here we will obtain an exact formula, containing both high-frequency and low-frequency components.

Starting from Hale's DMO operator Hale (1984)

 

$$\widetilde{\widetilde{P}}_0(\omega_0,k) = \int_{-\infty}^{\... ...\vert,k\right)\, \exp \left(i \omega_0\,t_1\,A\right) \,dt_1\;,$$ (74)

let us define its impulse response as a function G(t₀,t₁,y) such that

 

$$P_0(t_0,y_0) = \int\!\!\int P^{(0)}_1(t_1,y_0-\xi)\,G(t_0,t_1,\xi)\,dt_1\,d\xi\;.$$ (75)

According to this definition, the impulse response of operator (77) can be expressed as

 

$$G = {1 \over (2\,\pi)^2}\,\int\!\int A^{-1}\,\widetilde{P}^{... ... \omega_0\,(t_1\,A -t_0)\right)\, \exp (iky) \,d\omega_0\,dk\;.$$ (76)

Recalling the definition of Hale's factor A (19) and changing the order of integration in the double integral, we can then rewrite expression (79) to the form

 

$$G = {1 \over (2\,\pi)^2}\,\int_{-\infty}^{\infty} \exp(- i \... ...\right)} \over \sqrt{k^2+a^2}}\, \exp (iky) \,dk\,d\omega_0\;,$$ (77)

where $\hat{h}_1 = h_1\,\mbox{sign}(\omega_0)$, and $a={{\omega_0\,t_1} \over \hat{h}_1}$.

The inner integral in (80) is a known definite integral, evaluated explicitly in terms of cylinder functions Gradshtein and Ryzhik (1994). The idea of applying cylinder functions to the evaluation of the DMO impulse response was used previously by Berg 1985 and Hale 1991a. After evaluation of the inner integral, expression (80) transforms to

 

$$G = {i \over {2\,\pi}}\,\int_{-\infty}^{\infty} \exp(- i \om... ...0)\,{a \over 2}\, Z\left(a\,\sqrt{h^2-y^2}\right)\,d\omega_0\;,$$ (78)

where

 

$$Z(x) = \left\{ \begin{array} {lcr}\displaystyle{ H_0^{(1)}... ...x)+ iY_0(x)} & \mbox{for} & \omega_0 < 0 \end{array}\right.\;,$$ (79)

H₀⁽¹⁾ and H₀⁽²⁾ are Hankel functions of zero order, and J₀ and Y₀ are, respectively, Bessel function and Weber function of zero order. Separating the real and the imaginary parts of the integrand transforms expression (81) to the form

 

$$\left. +\int_0^{\infty} \sin(\omega_0\,t_0)\, Y_0\left(\omega_0\,\hat{\theta}\right)\, d\omega_0 \right]\;,$$ (80)

where

 

$$\hat{\theta} = t_1\,\sqrt{1-{y^2 \over h_1^2}}\;.$$ (81)

Note that the Bessel function in the first term of formula (83) follows from the real part of the Hankel function, which is in turn connected with the imaginary part of Hale's operator (77). For the same reason, the second term in formula (83) is the contribution from the real part of operator (77). Both definite integrals in (83) have known analytical expressions listed in integral tables Gradshtein and Ryzhik (1994). With the help of these expressions, the first term is expressed as

 

$$G^1(t_0,t_1,y) = -{t_1 \over {\pi\,h_1}}\, {\partial \over \... ...t{\theta}^2-t_0^2) \over \sqrt{\hat{\theta}^2-t_0^2}\right)\;,$$ (82)

while the second term transforms to

 

$$G^2(t_0,t_1,y) = -{t_1 \over {\pi^2\,h_1}}\, {\partial \over... ...^2}} & \mbox{for} & t_0 \gt \hat{\theta} \end{array}\right.\;.$$ (83)

The first term (G¹) is discontinuous on the line $t_0=\hat{\theta}$, which is the known form of the DMO impulse response Deregowski and Rocca (1981), while the second term (G²) is continuous and smooth everywhere. In order to prove this fact, one can easily verify that both the left-sided and right-sided limits of expression (86) for t₀ approaching $\hat{\theta}$ are

 

$$\lim_{t_0 \rightarrow \hat{\theta}-0} G^2= \lim_{t_0 \rightarrow \hat{\theta}+0} G^2= -{t_1 \over {3\,\pi^2\,h_1\,t_0^2}}\;,$$ (84)

and the limits of its derivative are

 

$$\lim_{t_0 \rightarrow \hat{\theta}-0} {\partial G^2 \over \... ...\over \partial t_0} = {4\,t_1 \over {15\,\pi^2\,h_1\,t_0^3}}\;.$$ (85)

The second (smooth) term of the impulse response, which comes from the real part of Hale's DMO, obviously does not contribute to the imaging properties of DMO. Moreover, it continues non-causally (and with monotonical growth) to the negative times (Figure 7), which contradicts the sense of DMO (and offset continuation) as an operator defined for positive times only. The energy of the second term is almost negligible, especially with respect to the high-frequency asymptotics. Therefore, in practice its presence doesn't affect the DMO behavior much. The conclusion that is made in this Appendix justifies the absence of this term in the DMO operator derived from the amplitude-preserving offset continuation (17).

 

offimp
Figure 7 Theoretical impulse response of Hale's DMO. Top: impulse response of the imaginary part; bottom: impulse response of the real part. Note the scale difference.