98 P. VALTR, P. PECHAČ, TROPOSPHERIC REFRACTION MODELING USING RAY-TRACING AND PARABOLIC EQUATION Tropospheric Refraction Modeling Using Ray-Tracing and Parabolic Equation Pavel VALTR, Pavel PECHAČ Dept. of Electromagnetic Field, Czech Technical University in Prague, Technická 2, 166 27 Praha 6, Czech Republic [email protected], [email protected] Abstract. Refraction phenomena that occur in the lower proper method and its implementation for a specific appli- atmosphere significantly influence the performance of cation. At the end a method for angle-of-arrival spectra wireless communication systems. This paper provides an calculation is presented for precise multipath propagation overview of corresponding computational methods. Basic simulations. properties of the lower atmosphere are mentioned. Practi- cal guidelines for radiowave propagation modeling in the lower atmosphere using ray-tracing and parabolic equa- 2. Radio Refractive Index tion methods are given. In addition, a calculation of angle- of-arrival spectra is introduced for multipath propagation The troposphere forms the lowest part of the atmo- simulations. sphere from the surface of the earth up to several km. From the propagation point of view, the troposphere is charac- terized by a refractive index, whereas the rate of the change of the refractive index with height is of crucial importance. Keywords The refractive index itself depends on absolute tempera- ture, atmospheric pressure and partial pressure due to water Radiowave propagation, Tropospheric refraction, vapor [1]. The predominant dependence of these quantities Ray-tracing, Parabolic equation. on elevation makes the troposphere a mostly horizontally stratified media. The refractive properties of air can be expressed in terms of the refractive index n or refractivity 1. Introduction N, where Long-range electromagnetic wave propagation in N = (n −1)⋅106 . (1) near-horizon direction is largely governed by spatial distri- bution of the refractive index in the atmosphere. Con- The refractive index of air at the surface of the earth is sideration of refractive properties of the lower atmosphere approximately 1.0003. Standard atmosphere is represented is thus of certain importance when planning and designing by an approximately linear decrease of refractivity at low terrestrial communication systems mainly because of mul- altitudes with a long-term mean value of the refractivity tipath fading and interference effects due to trans-horizon gradient equal to –40 N/km, [2], [3]. Radiowaves are bent propagation. Multipath phenomena can also be used for in consequence of a non-constant refractive index. The remote sensing applications. effect of the refractivity gradient to wave bending can be expressed using the radius of curvature of a ray repre- Recent propagation modeling methods considering senting the electromagnetic wave. The radius of curvature the refractive properties of the atmosphere employ ray- ρ of the ray can be well approximated by [4] tracing and parabolic equation approach. Ray-tracing is a 1 geometrical optics method while the parabolic equation ρ = − . (2) method is a full-wave approach to a homogeneous wave dn dz equation solution. Both methods have been known for many years, but new applications place new requirements The radius of the earth curvature is ρe = 6378 km. The on their implementation: multipath phenomena precise radius of the curvature of the ray under standard gradient is modeling, horizontally inhomogeneous troposphere, etc. higher than the radius of the earth curvature; both radiuses are equal for dN/dz = -157 N/km. The definition of This paper provides a basic description of methods modified refractivity and modified refractive index comes used in radiowave propagation prediction, taking into ac- from count the refractive conditions of atmosphere. Ray-tracing M = N +157z , M = (m −1)⋅106 (3) and parabolic equation methods as the most widely used techniques are addressed including implementation issues. where M and m are modified refractivity and modified Practical guidelines are given to enable the selection of a refractive index, respectively, and z is height in km. RADIOENGINEERING, VOL. 14, NO. 4, DECEMBER 2005 99 The effects of various refractivity gradients can be front at each point. Considering a high frequency harmonic seen in Fig. 1, where bending of rays representing radio field in inhomogeneous media and assuming small varia- waves is shown relative to earth curvature. The rays propa- tions of field intensity amplitude compared to wavelength gating under positive refractivity gradients are bent up- (large wavenumber k0) leads to the following pair of equa- wards. The standard refractivity gradient causes rays to tions determining the ray trajectory in two dimensions [6] bend downwards, but the curvature of the earth (1/ρe) ex- d ⎛ dx ⎞ ∂n(z) d ⎛ dz ⎞ ∂n(z) ceeds the curvature of the ray (1/ρ), which prevents trans- ⎜n(z) ⎟ = , ⎜n(z) ⎟ = (4) horizon propagation and creates a shadow area behind the ds ⎝ ds ⎠ ∂x ds ⎝ ds ⎠ ∂z radio horizon range. Ray propagating under a refractivity where only the dependence of refractive index on height z gradient equal to dN/dz = -157 N/km is exactly parallel to is considered and where s represents the length of the arc the surface of the earth. Gradients of less than –157 N/km of the ray and x denotes the horizontal distance. The left produce ducting where the curvature of the rays exceeds term of the first equation equals to zero causing the the curvature of the earth and the wave travels for a very bracketed term to be constant long distance behind the radio horizon. dx n(z) = n()z cosθ = C (5) dN/dh>0 ds where C is a constant and θ is the angle of the ray from dN/dh=-40 N/km horizontal direction. Eq.(5) represents Snell’s law. Con- dM/dh=117 M/km sidering the piece-wise linear profile in Fig. 3(a), the de- dN/dh=-157 N/km pendence of the refractive index in the horizontal segment dM/dh=0 M/km is in the form n2 = n1 + δ (z2 − z1 ) (6a) dN/dh<-157 N/km where δ is the refractive index gradient. Inserting (6a) into Snell’s law, the following set of equations determining the Fig. 1. Rays under various refractivity gradients. trajectory of the ray in one of the linear segments can be derived [7] Basically, there are two effects that can break the standard (6b) situation of the constant gradient of refractivity. The first is x2 = x1 + (α 2 −α1 ) δ an abrupt decrease of water vapor pressure with height, α 2 =α 2 + 2(n − n ) (6c) which occurs mostly in a narrow layer over water surface 2 1 2 1 and results in a so called evaporation duct, Fig. 2(a). The other is an inverse increase of temperature with height z z causing a surface or elevated duct, Fig. 2(b,c). Range of z1 z1 α1 heights of the ducting layer is determined by the top of z z 2 2 α negative M gradient layer and height of equal M below [5]. 2 Surface-based Height Height Height Elevated-duct Surface duct duct n n n x x x 2 1 1 2 (a) (b) Fig. 3. Refractive index profile and bending of a ray Eq. (6b) represents the dependence of x on the respective Refractivity, M units Refractivity, M units Refractivity, M units angle α2, which is in turn determined by the initial angle α1 (a) (b) (c) (6c). If the turn-around point of the ray exists within one Fig. 2. Types of ducts. layer as in Fig. 3(b), the two particular paths of the ray have to be treated separately. Possible ground reflections of the ray have to be considered. 3. Geometrical Optics In the formulas above, a rectangular coordinate sys- tem rather than a spherical one was considered. To be able Geometrical optics is a method suitable for the treat- to work with the flat earth model, the refractive index n has ment of propagation problems in homogenous media or in to be replaced by a modified refractive index m [7]. Fig. 4 slowly varying media compared to wavelength. The propa- shows the case of a standard atmosphere. Ray paths ob- gating field is locally considered a plane wave represented tained using (6a-c) are shown with the rays bent upwards by rays. Rays form trajectories perpendicular to the wave under earth flattening transformation. Fig. 5 shows an 100 P. VALTR, P. PECHAČ, TROPOSPHERIC REFRACTION MODELING USING RAY-TRACING AND PARABOLIC EQUATION example of a refractivity profile with the ducting layer received power prediction in a terrestrial point-to-point extending from 50 to 70 m. The tropospheric waveguide radio link, radar-cross-section computation and other re- effect causing the wave to travel far beyond the horizon lated propagation problems, the area of interest is within can be seen. several degrees of the preferred direction. In such a case a reduced wave equation of the parabolic type can be han- Although ray-tracing provides a rather simple tool for dled to obtain a computationally effective solution. finding the path of individual rays, problems may arise when treating diffraction effects caused by terrain irregu- A two-dimensional scalar wave equation in the larity and earth curvature. In such a case the principles of Cartesian coordinate system can be written as diffraction theory must be employed. Furthermore, a single 2 2 ray carries no amplitude information, and amplitude is ∂ ψ ∂ ψ 2 2 (7) 2 + 2 + k0 n ψ = 0 derived from the cross section of a tube of rays formed by ∂x ∂z several rays. In addition, geometrical optics does not pro- where ψ is the electric or magnetic field component, k0 is vide correct results at a ray caustic, i.e. at a locus of zero the wavenumber in a vacuum, n is the refractive index, x cross-section of the ray tube.