Yer dalga iletiminde parabolik denklem (PD) yöntemi
Parabolic equation (Pe) method in ground wave propagation
- Tez No: 46615
- Danışmanlar: Y.DOÇ.DR. LEVENT SEVGİ
- Tez Türü: Yüksek Lisans
- Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
- Anahtar Kelimeler: Dalga yayılması, Kırılma indisi, Parabolik denklemler, Wave propagation, Refractive index, Parabolic equations
- Yıl: 1995
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Bu çalışmada, kırılma indisinin enine ve boyuna iki iki boyutlu değiştiği ortamlarda dalga iletimi problemi Parabolik Denklem (PD) yöntemi ile ince lenmiştir. Öncelikle yüksek ve çok yüksek frekanslarda küresel bir zemin üzerine düşey olarak yerleştirilen kısa bir dipolden çıkan elektromagnetik dalgaların ya yılma özellikleri ve davranışları incelenmiştir. Ortam parametreleri kontuna bağlı olduğundan problemler basit analitik bi çimde ortaya konamazlar. Bu nedenle yaklaşık ya da salt sayısal yöntemlerle çözülmeye çalışılırlar. Dalga iletim problemlerinde, analitik - sayısal yaklaşık çözümlerden en yaygın olarak kullanılanlarından biri PD yöntemidir. Burada, öncelikle yöntem tüm dikkat edilmesi gereken noktalarının üzerinde durularak incelenmiş, daha sonra tam çözümleri bilinen kanonik bir yapı ele alınarak bu tam çözümlerle PD çözümleri karşılaştırılmıştır. Son olarak da, tam çözümleri bilinmeyen problemlere PD uygulanarak elde edilen sonuçlar yorumlanmıştır. Bu amaçlarla, çalışmanın bütününde olabildiğince çok değişik parametre takımları ile incelemeler yapılmış ve elde edilen tüm sonuçlar grafiklerle ve açıklamalarla verilmiştir. iv
Özet (Çeviri)
SUMMARY PARABOLIC EQUATION (PE) METHOD IN GROUND WAVE PROPAGATION In this thesis, the propagation phenomena in 2D complex environments with arbitrary transverse and slow longitudinal variations is described in terms of Parabolic Equation (PE) method. The model presented here represents propa gation of electromagnetic waves over a spherical, finitely conducting earth and allows specification of frequency, polarization, antenna pattern, antenna altitude and tilt angle. The well-known solution method of the parabolic form of the wave equation, split-step PE (SSPE) is compared with the reference solutions obtained from Helmholtz wave equation where analytic solutions exist and also with each other through surface duct-to-elevated duct transition to show the transformation of the surface trapped modes into the elevated beams. Propagation in complex waveguide environments which may involve bound aries and/or transversely confining refractive index variations are becoming more and more interesting for the electromagnetics and microwave society. Such prob lems can be solved with analytical methods which are mostly based on mode and/or ray summations or their hybrid combination. Analytical solutions are very instructive since they give a good insight into the physics associated with non-separable problems. However, they exist only for a small amount of ideal problems which are mostly non-physical, for this reason, a lot of effort have been given on finding approximate mode-like wave objects to built fields in non- separable environments. PE method combines the physical insight of the analytical solutions and the generality of numerical solutions in a computationally efficient scheme[8,9]. The PE method can be used to solve wave propagation problems everywhere, where in the variation of the geometry and/or medium parameters can be arbitrary in the transverse domain but is slow in the longitudinal direction so as to allow approximate reduction of the wave equation to the parabolic form. The parabolic form of the wave equation is very suitable for the numerical computations. Once the initial transverse field profile is given, SSPE algorithm, which is built to satisfy the transverse boundary conditions automatically, computes the transverse field distributions at any longitudinal range value proceeding step by step. At each range step the fast Fourier transform (FFT) routine is used to carry out thetransformations between the spatial transverse coordinate and transverse wave number coordinate. SSPE method has been applied to many important and interesting wave problems also including the longitudinally varying guiding media. In this study, first the complete derivation of the parabolic wave equation from the vector wave equation is presented[7,10]. The assumed spherical earth geometry is shown in figure 1. We are concerned with describing the propagation from a source located at 9 = 0 and r = rs in the region r > a (where a is the earth's radius) and in the far field of the source antenna. The emphasis in the following discussion is on the case where the source is a vertical electric dipole (VED), but the corresponding results for the horizontal electric dipole (HED) may be provided at various point. Source Location short dipole e0,a= 0 >M0 P(r,e><P) Figure 1. Earth - centered spherical geometry Assuming a time dependence of e“”' in a source-free medium, Maxwell's curl equation for the electric field E and the magnetic field H are easily combined to obtain the following vector wave equation for H ; V x Vx H -- xyxS -/zaj2 H= 0 (D where oj is the angular frequency, ft is the magnetic permeability (which has been assumed to be constant, [i - fio) and s is the electric permittivity. In the VED case, the non zero components of the fields are Er, Eg and H^;“with H = Hve^. VIAll quantities are assumed to be independent of ip because of the azimuthal symmetry of the source. Ignoring any tp dependencies, the scalar wave equation for Hv is will become r e + -^de 1 e sin 9 de{s\rx9H^) + fiıJH^ = 0 (2) The next step is to determine the substitution for Hv that will result in a relatively simple wave equation. This will be accomplished by requiring that the simplified wave equation will be in form of a Helmholtz equation. Using the following substitution ”.^vS-“^ (3) in (2) gives the following two-dimensional Helmholtz wave equation (d'r+i^+^y+^g-^+k^y=o (4.a) g = -V7^nl(d? + \%y l r2 ”'y/eââÖ (4.b) where the wave number is k = w/c, and the refractive index is, n = \J \ie\ ' \iqSq. It is convenient at this point to transform the problem into a rectangular co ordinate system in which the z axis corresponds to the surface of the earth with the origin directly below the source [3] on the surface. For near-horizontal propa gation, rapid fluctuations are factored out of the solution using the substitution u(z,x) = etkzU{z,x) (5) At this point, the parabolic (or paraxial) approximation is made to reduce the wave equation to an equation that is amenable to numerical solutions that march in range. The simplest approach is to assume that \&lu\ < k\dzu (6) vnand thus neglect the d2u term. The final form of the parabolic wave equation which will be used here can be expressed as [13] dl + 2ikdz + k2(n2-l + ^) u = o (7) The term 2x/a accounts for the spherical earth geometry; when it is neglected, (7) describes propagation over a fiat earth. Given an initial field and the appropriate transverse boundary conditions, eq.(7) is a well-posed initial boundary value problem which can be handled with longitudinaly marching computations in the form U(z, x) = exp.k ? i-m{z-z0) r -1 exp Mz-zo) 2k F[U(so,*)] (S) where m = n2 - 1 + 2x/a and Az = z - z0 represents the range step size along (longitudinal) propagation direction, J- and T~x represent Fourier and inverse Fourier transformations, respectively. Eq.(8) is the split-step marching solution of the parabolic equation (SSPE) which can be evaluated very efficiently utilizing FFT routines [12,14]. The SSPE routine is a numerically efficient algorithm for one way propagation problems which is easy to implement since it only needs the initial transverse field profile. But there are some important points which may be very tricky if one is not fully aware of the underlying mechanisms. First of all, the boundary conditions must automatically be satisfied as the field propagates longitudinally because the SSPE in eq.(8) does not account for the boundary conditions. Fortunatelly, most of the boundary conditions encountered in the propagation of the scalar fields are of Dirichlet and/or Neumann type and they can be handled by Sine and/or Cosine FFT. In practice this can be done by odd or even symmetric continuation of the initial transverse profile above and below the boundaries until the field decays exponently to zero. If this is not satisfied, an artificial absorbing layer is introduced to higher altitudes of interest to overcome the alising problem. The other important point is the determination of the necessary longitudinal step size. It should be chosen optimally to trace the longitudinal phase variation with minimum computation time. Since the linear negative gradient refractive index variation with respect to altitude is commonly used for tropospheric propagation, the inhomogeneous guid ing environment is characterized by the refractive index profile given as n2{x) = l-a0x N(x) = (n{x) - 1)106 (9) V1Uwhere n(x) and N(x) are the refractive index and the refractivity and x is the altitude. This kind of linear profile models the surface trapped propagation with an arbitrary positive constant, a0, controlling the height of the surface duct. Assuming time harmonic dependence as e8u,t, the 2D wave equation for a scalar field constituent U(x, z) function can be expressed as [d2x + d2z + k20n2(x)]U(x1z) = 0 (10) The finite conductivity of the earth's surface is characterized with the impedance type boundary condition [1 1] dxU(x, z) + aU(x, z) = 0 at i = 0 (11. a) Here a is the impedance parameter and U is subject to radiation conditions tf(x,*)Uoo- >0 (ll.b) U(x,z)\z^±oo^0 (ll.c) The location of the boundaries and the transverse refractive index variation of the medium permits the seperation of the 2D wave equation into transverse and longitudinal components which are ID differantial equations. The longitudinal equation gives a phase variation of exp(- ifiz) with exp(iu}t)time dependence, while transverse equation reduces to * + k2n2(x)-Ş2 u(x) = 0 u(x, z) = ü{x)t-lfiz { 12) _dx* Defining new transverse variables as k2n2{x) - f = Ax + B; p = -A~2/:i{Ax + B) ( 13) A = -a0k2 B = k2Q- &2, yields d2 - -P dp ü(p) = 0 (14) IXdifferent ranges and/or altitudes. The PD program uses FFT and inverse FFT modules at each longitudinal computation step. Extensive numerical tests have been performed for convergence, and for minimization of the anti-alising effects and truncation errors. Here, examples are presented wherein SSPE solutions on a surface duct and surface-to-elevated duct transition problems are compared with reference solu tions. The better critical transition is characterized by the transverse and lon gitudinal refractivity variations of the environment. The calculations and com parisons can be made at all radiowave frequency ranges but the results are given for the lower half (i.e., typically 3 - 15 MHz.) of the HF band. These results are applicable to surface wave HF communication especially for the surface wave HF radar applications. In the computations the coverage of the altitude and range extends to 3km. and 200km., respectively, which is the typical region of interest in surface wave HF communication systems. The initial refractivity gradient for the model is chosen as dN/dh = iOONunits/km which is associated with at mospheric ducts ör trapping layers. The positive constant ao which controls the surface duct height will then be 4. 10-7. n(x,z) i Z = -100 z = 0 z = 100 Range (km) Figure 2. The structure of the problem and the refractive index variations at different observation ranges SSPE algorithm is first tested on longitudinally invariant surface duct defined by the first profile in fig. 2. Assuming that the initial 400 N units /km. transverse refractivity gradient remains invariant for all observation ranges, yields an exact solution in terms of Normal Modes (NM) [20]. The first comparison is carried out between the NM(exact solution) and PE computations. Fig. 3. shows various altitude profiles at different observation ranges. The initial profile is built with the superposition of the equally excited first ten trapped modes and then fed into the PD.FOR algorithm. The truncation of the spectrum in kx domain is chosen to cover the transverse wave number of the highest NM considered in the altitude field distribution for the SSPE computations. The loss factor of a is introduced XIbetween Xmax/2 and Xmax. A very good agreement between the NM and SSPE results is clearly seen from the figure for the altitudes extending from the surface to five kilometers. 10000 7600 - ^-»., 2500- Figure 3. The comparisons of the exact analytical results(NM) with the SSPE compu tations at different ranges (a: loss factor) In the PD.FOR algorithm, the initial transverse profile can be constructed to correspond to the radiation field of specific antenna isotropic in the horizontal plane. In order to compare the SSPE method with NM results, the initial trans verse profile of the antenna pattern is simulated with the superposition of the NM with the suitably chosen excitation coefficients. The radiation pattern in the elevation plane simulating the Log-Periodic dipol antenna for 3MHz. is apporxi- mated with the superposition of suitably excited the first ten NM assumed which are then fed into SSPE algorithm. The results of the NM and SSPE computations for the transverse field profiles at different observation ranges are given in fig. 4. An excellent agreement between the two methods is clearly seen in the figure for the Dirichlet type boundary condition. Finally, the surface duct-to-elevated duct transition is examined. There is no reference solution for the surface duct-to-elevated duct transition. Therefore, only the SSPE results corresponding to this transition are given Fig. 5. The initial transverse field distribution at the begining of the transition region (c = Z\) is again calculated via LINEER.FOR as a modal superposition over the modes of the homogeneous surface duct. In fig. 5., the first two modes are considered in order to be able to demonstrate the transformation of energy from a trapped mode to the radiating beam. This transformation is clearly seen from the figure. In this study, a powerful tool for one way propagation problems is examined on a canonical surface duct as well as on a complex surface duct-to-etevated duct transition problems. The SSPE algorithm in this form may be used for all kinds of refractive index profiles under slow longitudinal variations or in the paraxial regions and for any transverse boundary variations [18,19]. xn(a) 6000 4000- 2000- (b) Figure 4. The comparisons of the analytical solution based on modal summation and SSPE algorithm. Figure 5. The transformation of the surface trapped modes into elevated beams for the lowest two modes. xiu
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