Is it possible to measure with great precision the time
at which a very short EM pulse reaches a receiver situated
at some distance from the emitter?
The time should be recorded on the emitter's clock and
also on the receiver's clock.
is the  receiver situated
 at some distance from the emitter with a rough terrain or or smooth
you can use this formula if you modulate two separate frequecies of
coherent light with an Em frequency
over any terrain at any distance
∇′2ψ(x′, z′, ω) + k2(ω)n2ψ(x′, z′, ω) = 0 (1)
where ψ represents the field for either the vertically or horizontally
polarized wave. For
vertically polarized wave, ψ is the magnetic field, which has
component only along the
y direction. For horizontally polarized wave, ψ represents the
electric field, which is
pointed along the y direction. In addition, ∇′2 = ∂2/∂x′2 + ∂2/∂z′2,
k2(ω) = ω2µ0ǫ0,
and n is the index of refraction of the propagating medium. It can be
shown that
many important radio wave propagation phenomena can be reduced to this
twodimensional
problem [3]. In two dimensions, the irregular terrain is described by
the function z′ = f(x′). In this paper the terrain surface profile
f(x) is assumed to
be a stochastic process. The problem we are concerned with is that of
radio wave
propagation over irregular terrain, in which a transmitter located at
the horizontal
position x′ = x0 and at a height of h above ground radiates a
transient pulse. The
pulse then propagates in the positive x′ direction until it reaches a
receiver located
at the horizontal position x′ = x, and a height of z above ground. The
geometry of
the problem under consideration is shown in figure 1, where the
horizontal distance
between the transmitter and receiver is R.
We further assume that the terrain material can be approximated by
perfect
electric conductors (PEC). Therefore, for vertically polarized wave,
equation (1)
satisfies the Neumann boundary condition,
∂ψ
∂z′   z′=f(x′)
= 0. (2)
While for horizontally polarized wave, the boundary condition is given
by the Dirichlet
boundary condition,
ψ|z′=f(x′) = 0. (3)
We make the assumption that the terrain elevation varies on a scale
length large
compare to the wavelength of the radio wave and also that the wave
propagates at small
grazing angle relative to the x axis. Then, using the forward
scattering approximation,