The definition of *k*_{z} as obscures two aspects of *k*_{z}.
First, which of the two square roots is intended,
and second, what happens when .For both coding and theoretical work we
need a definition of *ik*_{z} that is valid
for both positive and negative values of and for all *k*_{x}.
Define a function by

(17) |

Let us see why is positive
for all real values of and *k*_{x}.
Recall that for ranging between , rotates around the unit circle
in the complex plane.
Examine Figure 10
which shows the complex functions:

- 1.
- ,
- 2.
- ,
- 3.
- ,
- 4.
- , and
- 5.

Figure 10

The first two panels are explained by the first two functions. The first two functions and the first two panels look different but they become the same in the practical limit of and .The left panel represents a time derivative in continuous time, and the second panel likewise in sampled time is for a ``causal finite-difference operator'' representing a time derivative. Notice that the graphs look the same near .As we sample seismic data with increasing density, ,the frequency content shifts further away from the Nyquist frequency. Measuring in radians/sample, in the limit , the physical energy is all near .

The third panel in Figure 10
shows which is a cardioid that
wraps itself close up to the negative imaginary axis without touching it.
(To understand the shape near the origin, think about the square
of the leftmost plane. You may have seen examples
of the negative imaginary axis being a branch cut.)
In the fourth panel a small positive quantity *k*_{x}^{2} is added which
shifts the cardioid to the right a bit.
Taking the square root gives the last panel
which shows the curve in the right half plane
thus proving
the important result we need,
that for all real .It is also positive for all real *k*_{x} because
any *k*_{x}^{2}>0 shifts the cardioid to the right.
The additional issue of time causality in forward modeling
is covered in IEI.

Luckily the Fortran `csqrt()` function assumes the phase of
the argument is between exactly as we need here.
Thus the square root itself will have a phase between as we require.
In applications, would typically be chosen proportional to the maximum time on the data.
Thus the mathematical expression might be rendered in Fortran as
`cmplx(qi,-omega)`
where
`qi=1./tmax`
and the whole concept implemented as in function `eiktau()` .
Do not set `qi=0` because then the `csqrt()` function
cannot decipher positive from negative frequencies.

complex function eiktau( dt, w, vkx, qi ) real dt, w, vkx, qi eiktau = cexp( - dt * csqrt( cmplx( qi, -w) ** 2 + vkx * vkx /4. ) ) return; end

Finally, you might ask, why bother with all this careful theory
connected with the damped square root.
Why not simply abandon the evanescent waves
as done by the ```if`'' statement in subroutines
`phasemig()` and `phasemod()`?
There are several reasons:

- 1.
- The exploding reflector concept fails for evanescent waves
(when ).
Realistic modeling would have them damping with depth.
Rather than trying to handle them correctly we will make a choice,
either (1) to abandon evanescent waves effectively setting them to zero,
or (2) we will take them to be damping.
(You might notice that when we switch from downgoing to upgoing,
a damping exponential switches to a growing exponential,
but when we consider the adjoint of applying a damped exponential,
that adjoint is also a damped exponential.)
I'm not sure if there is a practical difference between choosing to damp evanescent waves or simply to set them to zero, but there should be a noticable difference on synthetic data: When a Fourier-domain amplitude drops abruptly from unity to zero, we can expect a time-domain signal that spreads widely on the time axis, perhaps dropping off slowly as 1/

*t*. We can expect a more concentrated pulse if we include the evanescent energy, even though it is small. I predict the following behavior: Take an impulse; diffract it and then migrate it. When evanescent waves have been truncated, I predict the impulse is turned into a ``butterfly'' whose wings are at the hyperbola asymptote. Damping the evanescent waves, I predict, gives us more of a ``rounded'' impulse. - 2.
- In a later chapter we will handle the
*x*-axis by finite differencing (so that we can handle*v*(*x*). There a stability problem will develop unless we begin from careful definitions as we are doing here. - 3.
- Seismic theory includes an abstract mathematical concept known as branch-line integrals. Such theory is most easily understood beginning from here.

12/26/2000