MODELING RADAR PROPAGATION OVER TERRAIN Modeling Radar Propagation Over Terrain Denis J. Donohue and James R. Kuttler M athematical techniques are presented for modeling electromagnetic propaga- tion over an irregular boundary. The techniques are aimed at upgrading the Applied Physics Laboratory’s Tropospheric Electromagnetic Parabolic Equation Routine (TEM- PER) model for radar propagation over terrain. The physical domain with an irregular boundary is mapped to a rectangular domain, where a numerical solution can be generated by the same approach used in TEMPER. This new method is applied to several model terrain problems and shown to be accurate and practical for a reasonable range of surface slopes. Interesting results are discussed for the shadowing of radar by terrain obstacles and the detection of low-flying targets over mountainous terrain. (Keywords: Electromagnetics, Ground clutter, Parabolic equation, Propagation, Rough boundaries.) INTRODUCTION The propagation of radar waves at low grazing angles interaction with the ground occurs. The Theater Sys- over terrain is a critical area for numerical modeling tems Development Group of APL’s Air Defense and performance prediction. A principal goal is esti- Systems Department has developed the computational mating ground clutter, or surface backscatter, an model TEMPER (Tropospheric Electromagnetic Para- obvious impediment to target detection. In addition, bolic Equation Routine) to better understand and diffuse reflection from the ground can alter the coher- predict the effect of such environmental factors on ent interference between the direct and reflected radar system performance. beams, adding additional uncertainty to the radar’s TEMPER has been under development at APL since coverage pattern. These terrain effects become even the early 1980s. The first objective was to accurately more pronounced when coupled with atmospheric re- calculate electromagnetic propagation over the sea in fraction. The inhomogeneity of the atmosphere, which complicated refractive environments.1–5 TEMPER is often takes the form of horizontally stratified density currently used extensively on several Navy programs to layers, can redirect radar energy such that repeated provide propagation calculations for radar system de- JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 18, NUMBER 2 (1997) 279 D. J. DONOHUE AND J. R. KUTTLER sign studies, posttest reconstruction, and in situ ship- wavelength is particularly important to propagation board environmental assessment. Although TEMPER over the ocean surface and to certain types of terrain is mature and well established as a predictor of prop- as well (a related problem is scattering from ground agation over the sea, the Navy’s current emphasis on cover/foliage). Alternative techniques have been de- littoral operations results in a need for accurate prop- veloped to account for such features,4 and they are not agation calculations over irregular terrain as well. This discussed here. article describes recent efforts to develop algorithms to address this need. The Parabolic Wave Equation (PE) was first derived REVIEW OF PARABOLIC WAVE in the 1940s by Fock,6 but analytical solutions were EQUATION AND FOURIER/SPLIT- developed only for problems involving unrealistically STEP SOLUTION simple refractive conditions. In 1973, Hardin and Tap- TEMPER is based on the PE, an approximation to pert introduced an efficient numerical approach for 7 the reduced wave (or Helmholtz) equation, where it is solving the PE called the Fourier/split-step algorithm. assumed that the wave energy propagates predominant- This algorithm was used primarily for underwater prop- ly in the forward or horizontal direction. This approx- agation until 1981, at which time Harvey Ko and col- imation has been found to work quite well for modeling leagues in APL’s Submarine Technology Department low-grazing-angle radar propagation. The principal (STD) modified an acoustic model to address electro- 1 advantage of the PE approach is numerical efficiency. magnetic propagation in the troposphere. The new By neglecting backscattered wave energy and restrict- model was called the Electromagnetic Parabolic Equa- ing propagation to small angles with respect to the tion (EMPE). horizon, highly optimized numerical methods can be In 1984, Dan Dockery, then in the Fleet Systems employed to rapidly calculate propagation over tens or Department, working with the STD developers and even hundreds of kilometers in range. Altitudes are members of the APL Research Center, began modifi- generally limited to a few kilometers above the Earth’s cations to expand EMPE’s capabilities as a radar prop- 2–5 surface. The PE approach is also well suited to incor- agation prediction tool. These upgrades included in- porating atmospheric refraction. Rigorous solution over corporation of an impedance boundary condition, a a large-scale irregular boundary has previously been rough surface model, and flexible antenna pattern al- considered a limitation of the PE method. gorithms; increased numerical efficiency and robustness For this article, we consider a scalar form of the were also achieved. Eventually, the model was renamed Helmholtz wave equation, corresponding to one com- TEMPER to avoid confusion with the original program. ponent of a vector electric (or magnetic) field. Further- An extensive experimental campaign was also under- more, we assume azimuthal homogeneity of the atmo- taken between 1984 and 1989 to validate EMPE/TEM- sphere and terrain, so that solutions are generated in PER using calibrated propagation data collected in 2,3 a two-dimensional (range x vs. altitude z) slice. Under measured refractive conditions. These tests estab- these assumptions, the Helmholtz equation for the field lished the accuracy of the PE Fourier/split-step ap- f has the form proach for predicting radar propagation over the sea in complicated, range-varying, refractive environments. At present, however, TEMPER is limited in its ∂2∂2 ff(,)xz (,)xz 22 ability to rigorously account for irregular terrain. Ac- + +=k n(,)(,) xzf xz 0, (1) ∂ 2∂ 2 curate numerical solutions can only be guaranteed x z when propagating over mean smooth surfaces, such as a flat plane or spherical Earth. An approximate tech- where k = 2p/l is the wavenumber of the field and n nique, described in the next section, was previously is the index of refraction. It is important to emphasize introduced to account for arbitrary changes in surface that all of the effects of atmospheric refraction are slope. However, this technique introduces nonquanti- incorporated into n, and furthermore, the boundary fiable errors into the solution. This article describes mapping methods discussed in the next section result techniques for rigorously incorporating an irregular in modification to n only. This is critical to the ap- boundary into the TEMPER model. A principal re- proach, since it allows us to account for both atmo- quirement is that the new technique allows one to spheric and terrain effects through the standard meth- retain the Fourier/split-step numerical approach. We ods used to solve Eq. 1. emphasize that the terrain under consideration con- To obtain the parabolic form of the wave equation, tains roughness features or slope variations on scales we first assume that the field f propagates as time- that are much larger than the typical radar wavelength harmonic (eiwt), cylindrical (two-dimensional) waves of ⋅ l (centimeters to meters). The effect of fine-scale the form f(,)xz= uxze (,)i(k r) / r , where r is the dis- roughness that is comparable to or smaller than the tance from the source in the two-dimensional space. 280 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 18, NUMBER 2 (1997) MODELING RADAR PROPAGATION OVER TERRAIN The function u(x,z) can be regarded as the slowly satisfies Dirichlet’s boundary condition [u(x, 0) = 0], varying (in space) envelope of the propagating wave whereas the V-pol field satisfies Neumann’s boundary field. Our objective is a wave equation for u in which condition [∂u(x, 0)/∂z = 0]. These boundary conditions the traveling wave dependence is factored out of the are satisfied exactly by using either sine (Dirichlet or problem. By substituting u into Eq. 1 and dropping a H-pol) or cosine (Neumann or V-pol) transforms in Eq. term containing ∂2u(x, z)/∂x2 (the paraxial approxima- 3. For nonflat surfaces, the z derivative becomes a tion), we arrive at such a form, normal derivative, which couples both x and z. In that case, satisfying the boundary condition is considerably more involved. An approach discussed in the next ∂ ∂2 uxz(,)= i +−k 2 section is to map the irregular surface via coordinate [nxz (,)1 ] uxz (,). (2) ∂x 22k∂z2 transformation to a space in which the surface is locally flat, so that a similar splitting of cosine and sine trans- forms may be employed. We emphasize that once so- The paraxial approximation used to derive Eq. 2 as- lutions are obtained for the orthogonal H and V po- sumes that the propagation is nearly horizontal, and larizations, the solution for any linear polarization state that gradients in the horizontal direction are small may be obtained by simple linear combination. compared with the vertical. For a complete develop- The preceding discussion considers the solution of ment of this equation, the reader is referred to Ref. 4. the PE over surfaces that are flat or that may be made Equation 2 is a parabolic partial differential equation flat by simple coordinate transformation. Although the that can be solved as an initial value problem. A start- following section considers more general boundaries of ing field, u(z), is specified for an initial x, and the nonzero slope, the current TEMPER code uses an ap- solution is marched forward in range. The marching proximate technique for such boundaries that may be method is highly efficient numerically, particularly mentioned here. The terrain is approximated by a se- since the range and altitude stepping can be decoupled quence of up and down stair steps. At each down step, using the Fourier/split-step method.