Radio Science Bulletin 325 - June 2008 - URSI

anddQ 1 r ∂n= + n −Qdθ2 2 2n − Q ∂r2dr rQdθ =n − Q2 2where ( ) ( ) 2 2Q n dr dθdr dθr2 2(42), (43)= + . We now only needto solve two ordinary differential equations to find a ray, alot more efficient than the four equations for the planarHaselgrove equations. The downside is that we have ignoredlateral deviations and the background magnetic field. Somemeasure of the background magnetic field can beincorporated by ray tracing at effective wave frequenciesω± ω H 2 , where ω H is the gyro frequency at the midpointof the ray path (the different effective frequencies correspondto the two different roots of Equation (23)). (The readershould refer to [10] for more accurate expressions of effectivefrequency.) Once again, the Runge-Kutta-Fehlbergtechnique can be used to efficiently solve the ray-tracingequations.If we move to a nearby ray path, the deviations ( δ rand δ Q ) in quantities r and Q can be calculated fromand( δ2 2 2) δ 1 ∂ ∂d Q r ⎡ r⎛ n n ⎞= ⎢ +dθ2 2 2 r r 2n Q ⎢⎜ ∂ ∂r⎟− ⎣ ⎝ ⎠⎛ 2 21 ∂n 1⎞ ⎛ δ r ∂n⎤ ⎞ −⎜− ⎟ −QδQ⎥⎜2 22( n Q )∂r r ⎟⎜ 2 ∂r⎟⎥⎜ − ⎟⎝ ⎠⎝ ⎠ ⎥ ⎦( δ )d r r ⎡δrQ= δ Qdθ2 2⎢ +n − Q ⎣ r2Q ⎛δr ∂n⎞⎤QδQ2 2 ⎜2 ∂r⎟(44)− − ⎥ . (45)n − Q ⎝⎠⎦⎥Since power flows within the confines of the rays emanatingfrom a source, the above deviation equations are mostuseful in calculating the change in power density along a raypath. Lateral to the ray plane, the deviation can be estimatedby assuming the rays to follow great-circle paths throughthe origin of the main ray. However, this approximation isonly valid when the lateral variations in the ionosphere areweak.5. Analytic TechniquesConsider the variational principle for a twodimensionalray path in Cartesian coordinates ( xy , ). Forthe case where the refractive index depends only on the ycoordinate, the variational principle becomes12R ⎡ 2⎛dy⎞ ⎤δ n( y)⎢ + 1 ⎥ dx = 0 , (46)∫ ⎜ ⎟dxT⎢⎣⎝⎠ ⎥⎦from which a standard result of variational calculus yieldsa first integral of the Euler-Lagrange equations( )n y12⎡ 2⎛dy⎞ ⎤⎢ + 1 ⎥ = C , (47)⎜ ⎟⎢⎣⎝dx⎠ ⎥⎦where C is a constant of integration. This is Snell’s law fora horizontally stratified ionosphere, and is a single firstorderordinary differential equation for the ray path. In thecase of radial coordinates with the refractive index dependingon the radial coordinate r alone, there is a first integral of2the form () ( )n r r dθ ds = C . This is Bouger’s law, whichis the generalization of Snell’s law to a spherically stratifiedionosphere (see [11]). In [12], it was shown that importantquantities such as ground range could be derived analyticallyin the case of an ionospheric layer with refractive index2n = α + β r+ γ r for rb < r < rm + ym, and n = 0elsewhere. Parameters α , β , and γ are given by1 ( ) 2 ( )22α = − ωc ω + rbωc ymω, β = 2r b r m ω c ω , andγ = ( r ) 2mb r ωc ymω, where y m is the thickness of thelayer, r b is the radius of the base of the layer, r m is theradius of minimum refractive index, and ω c is the plasmafrequency at this height. Such an ionospheric layer is knownas a quasi-parabolic layer. It was shown in [12] that a raylaunched at the surface of the Earth with an initial elevationφ will land at a distance0⎡D = 2r E⎢ φ −φ0⎢⎣( )⎤⎥2rEcosφ0β − 4αγ⎥− ln⎥ ,(48)2 γ2⎛ 1 1 ⎞ ⎥4γ sinφ+ γ + β ⎥⎜rb2 γ ⎟⎝⎠ ⎥ ⎦where cosφ= rErbcosφ0, and r E is the radius of theEarth. Such results arguably provide the most efficient formof ray tracing.22TheRadio Science Bulletin No 325 (June 2008)