Statistical Estimation and Tracking of Refractivity from Radar Clutter

17
which results in the cancelation of the first term in (1.9). Therefore, to approximate
the wave equation in this parabolic equation form the following conditions must
be satisfied [14]:
1. The equation is only valid within a narrow beam geometry called the paraxial
cone, typically not more than 10 o . The first order error term associated with
increased propagation angle is proportional to sin 4 α, where α is the angle
between the propagation direction and the horizontal paraxial direction r.
This error term will increase with α as 10 −7 , 10 −3 , and over 10 −2 for 1 o , 10 o ,
and 20 o , respectively. More computationally expensive wider angle schemes
can be implemented using the Padé coefficients however lower atmospheric
duct calculations will typically require an angle less than 0.5 o (see Fig. 1.5)
making the fast narrow angle code preferable to wide angle finite difference
schemes.
2. The field is valid only in the far-field, not close to the source.
3. The medium is only weakly inhomogeneous such as the part-per-million
changes involved here.
4. Most of the energy should be propagating forward without any significant
back scattering.
Equation (1.17) can be marched using the split-step fast Fourier transform
(FFT) method, where u(r, z) and U(r, p) are Fourier transform pair related by
∫ zmax
U(r, p) = F {u(r, z)} = u(r, z)e −jpz dz (1.21)
−z max
u(r, z) = F −1 {U(r, p)} = 1 ∫ pmax
u(r, z)e jpz dp, (1.22)
2π −p max
where the transform variable is defined by p = k sin α, and the domain truncation
height is related to p max using the Nyquist criteria z max p max = Nπ, N being the
FFT size. Taking the FFT of the SPE (1.19) one can compute the closed form