Complex results in inverse Fourier transform for simulating ocean water

Question

I don't understand the equation37 in simulate ocean water by Jerry Tessendorf.The result is all complex number, how to be the slope.Even if I compute the magnitude of it，the result is just positive which is obvious wrong.As There must be some points whose slope is negative.Who can help me.Thank you.

http://www-evasion.imag.fr/Membres/Fabrice.Neyret/NaturalScenes/fluids/water/waves/fluids-nuages/waves/Jonathan/articlesCG/simulating-ocean-water-01.pdf

Answer 1

If $h(x)$ is a real valued function we have that since $h(x) = \bar{h}(x)$ that its Fourier series

$$ h(x) \approx \sum \tilde{h}(k) \exp ikx = \sum \bar{\tilde{h}}(k) \exp -ikx \approx \bar{h}(x)$$

So $\tilde{h}(k) = \bar{\tilde{h}}(-k)$. This is a fundamental property of the Fourier transform of real valued functions.

Now if we write

$$ \nabla h(x) \approx \sum i k \tilde{h}(k) \exp ikx = \sum \eta(k) \exp ikx $$

we note that

$$ \bar\eta(k) = \eta(-k) $$

by a direct computation. And hence

$$ \nabla h(x) = \overline{\nabla h}(x) $$

is a real valued function.