Mathematical space-time model of a sky wave radio field

Transcription

1 RADIO SCIENCE, VOL. 41,, doi: /2005rs003332, 2006 Mathematical space-time model of a sky wave radio field B. G. Barabashov, 1 M. M. Anishin, 1 and O. Y. Pelevin 1 Received 13 August 2005; revised 30 December 2005; accepted 26 January 2006; published 17 May [1] The three-dimensional mathematical field model of HF sky waves reflected by a spatially nonuniform nonstationary magnetoactive ionosphere is described. The model is based on the structural physical approach, which leads to complete understanding of the field structure in space and time associated with specific geomagnetic conditions. Citation: Barabashov, B. G., M. M. Anishin, and O. Y. Pelevin (2006), Mathematical space-time model of a sky wave radio field, Radio Sci., 41,, doi: /2005rs Introduction [2] In the design of HF communication and directionfinding systems, a sufficiently complete knowledge of the properties of sky wave fields is required in order to develop effective algorithms for space-time processing of signals coming from antenna arrays. Similar problems arise in modem technology design. The traditional approach of natural tests is hardly feasible for ionospheric channels. Here computer simulation can be a very valuable help. [3] The paper presents the three-dimensional mathematical model of the field of HF sky waves reflected by a spatially nonuniform nonstationary magnetoactive ionosphere. The model represents the further extension of the two-dimensional one [Barabashov and Vertogradov, 1996, 2000], and like the latter differs in principle from existing phenomenological ones [Goodman, 1992] in that it makes use of actual mechanisms of HF propagation in the time-dependent irregular ionosphere. The present model is based on a structural physical approach, which leads to complete understanding of field structure in space and time for specific geomagnetic conditions. Thus is accomplished the main requirement of the simulation: adequacy of the model. 2. Theory [4] Construction of the model is determined by two major moments which represent researchers latest opinions and are also confirmed by lots of experimental data. The first moment is that the problem of complete 1 Department of Radio Physics, Faculty of Physics, Rostov State University, Rostov-on-Don, Russia. Copyright 2006 by the American Geophysical Union /06/2005RS description of a sky wave field with all its peculiarities is totally solved by numeric simulation of ray paths in spatially nonuniform magnetoactive plasma. The second moment is that all main characteristics of sky waves can be interpreted in the first order of approximation within the model of medium- and large-scale traveling ionospheric disturbances Basic Points of the Model [5] 1. The total field at the reception site results from the interference of a few individual rays with different space- and time-dependent amplitudes and phases. [6] 2. Time variation of individual ray and total field parameters is attributed only to terminator travel and traveling ionospheric disturbances (TID). [7] 3. The ionosphere is taken as a three-dimensional nonuniform medium along a ray path, thus involving arbitrarily directed gradients. [8] 4. The multipath field is sought under geometrical optics approximation in the form of a solution of an eikonal equation. [9] Experimental validation of the basic points is as follows. [10] Discrete structure of the field is confirmed by numerous oblique ionograms. [11] Spectra of a multipath field obtained for s appear as discrete lines. Line spread for one-hop rays at oblique incidence, as a rule, does not exceed Hz. [12] For a fixed length path, a Doppler shift grows with the increase of operating frequency (that is, for a deeper penetration of wave into layer). Doppler shift drops with the increase of a path length. [13] Time dependency of Doppler shift and of angles of arrival (elevation and azimuth) has a form of an oscillating process with a quasiperiod from a few minutes up to a few hours. [14] Scattering mechanisms do not significantly affect field strength for one-hop paths. The values predicted for 1of6

2 no scattering differ from the experimental ones by 2 3 db on average. [15] An ample bibliography can be found in the work by Goodman [1992] Coordinate Systems [16] The model uses three coordinate systems. The first one is Cartesian with origin in the center of the Earth, the z axis pointing at the North Pole and the x axis lying in the plane of the Greenwich meridian. In this system a ray is described by spherical coordinates (j, q, r), where q counts from the z axis to radius vector r of the ray, j describes the direction of r with respect to the x axis in the oxy plane, and r is the length of r. Similar coordinates are used for a wave vector k =(k j, k q, k r ). The second system is Cartesian local centered at the transmission point with the z axis pointing upward, the y axis pointing to geographical north, and the x axis pointing eastward. Initial values of the wave vector are given by azimuth a 0 and elevation D 0. Azimuth a counts from the y axis, and elevation D starts at the oxy plane. Finally, the third coordinate system is Cartesian physical. It has the z axis in the direction of à wave vector, and the x and y axes are located such that major and minor semiaxes of polarization ellipse are oriented along these axes, respectively Model of Nonstationary Ionosphere Irregular in Three Dimensions [17] Spatial distribution of electronic density in the present model is set in nodes of a three-dimensional (3-D) grid for r, q, and j coordinates incremented by Dr, Dq, and Dj, respectively, with time increment Dt. The number of nodes is not large and is subject to external variation. In the nodes, the nondisturbed electronic density profile is described by the International Reference Ionosphere (IRI-2001) ionospheric model. The values thus obtained are then approximated by 3-D cubic spline, which provides continuity of the function and its derivatives. [18] In trial usage of the model the optimum spacetime grid increments were found as follows: Dr = 4 km, Dq = 0.2, Dj = 0.2, and Dt 5 min. Such values provide the desired accuracy of modeling at acceptable computational expense. [19] Wave disturbances (TID) are modeled by a packet of traveling monochromatic waves: Nr; ð q; j; tþ ¼ N 0 ðr; q; j; tþ 1 þ X3 d i cos 2p t T i¼1 i þ p ri ðr R 0 Þ þ p qi R 0 q þ p ji R 0 sinðþj q þ F 0i! ; where N 0 (r, q, j, t) is space-time distribution of ionization for nondisturbed ionosphere in altitude r, angle coordinates j and q, and time t; d i is a relative magnitude of TID having period of T i ; p ri, p ji, and p qi are radial, horizontal, and normal components of the TID wave vector; F 0i is the initial phase of the ith harmonic, and R 0 is the radius of the Earth. Components p ri, p ji, and p qi of the ith harmonic can be expressed in the form p ri ¼ 2p L i sin b i ; p ji ¼ 2p L i cos b i sin g i ; p qi ¼ 2p L i cos b i cos g i ; where L i is the wavelength, b i is elevation from the horizontal plane, and g i is azimuth, starting at the oy axis. Normally, n of 2 or 3 is enough. [20] Ionospheric absorption is determined using the collision frequency profile which does not vary along a path and is close to the gas kinetic profile (in MHz) vh ð Þ ¼ K½expð12:8076 0:158hÞ 0:0042hÞŠ; þexpð 6:175 where h is altitude in km. The possible variation of the v(h) profile due to geocyclic and heliocyclic conditions is simulated by a correction factor K, which is calculated using global empiric International Telecommunication Union recommendation (ITU-R) maps for 2.2 MHz absorption Ray Tracing [21] As is known, ray path deflections from the great circle plane (azimuth deflection) due to ionospheric anisotropy become significant (over 0.5 ) for traces shorter than km. If these deflections are neglected, then there is the possibility of performing ray tracing neglecting the effect of geomagnetic field. Thus the problem of multipath field simulation simplifies substantially, and the influence of the geomagnetic field on other characteristics of a ray such as field strength, elevation angles, group and phase delays, and Doppler frequency shifts can be represented approximately, which is sufficient for simulation accuracy. 2of6

3 [22] For isotropic ionosphere with three-dimensional irregularity, the set of characteristic equations derived from an eikonal equation takes the following form: dj dp 0 ¼ 1 r sin q k j dq dp 0 ¼ 1 r k q dr dp 0 ¼ k r dk j dp 0 ¼ 1 r sin q dk q dp 0 ¼ 1 r dk r dp 0 ¼ 2 þ k q ddf dp 0 ¼ f dm 2 2c dt dp dp 0 ¼ m2 k j sin q dr dp 0 rk j cos q dq dp k dr q dp 0 þ rk j cos q dj dp 0 dq dp 0 þ k j sin q dj dp 0 ð1þ [23] Here D and a are ray coordinates (elevation and azimuth angles, respectively), and group path P 0 is chosen to parameterize ray trajectory. The other variables denote the following: P is phase path; (k r, k q, k j ) are physical polar coordinates of wave vector k; df is Doppler frequency shift, m 2 = 1 (f n /f) 2 ; l is the wavelength; and f = c/l and f n are the operating and plasma frequencies, respectively. [24] For set (1), initial conditions at ground point with coordinates (j 0, q 0, R 0 ) are given as follows: j ¼ j 0 q ¼ q 0 r ¼ R 0 k j ¼ cos D 0 sin a 0 k q ¼ cos D 0 cos a 0 k r ¼ sin D 0 : ð2þ [25] Set (1) is solved using the fifth-order Runge- Kutta-Falberg method with variable step, which allows automatic selection and adjustment of the integration step in the ray-tracing process Solution of Boundary Value Problem [26] For a given trace, to solve the point-to-point problem in the 3-D ionosphere is to find initial azimuth a 0 and elevation D 0 such that a ray reflected from the ionosphere arrives exactly at the reception site (q k, j k ). Thus we have a set of two nonlinear equations in independent variables a and D: qðd; aþ q k ¼ 0 jðd; aþ j k ¼ 0: ð3þ [27] Here q(d, a) and j(d, a) are current coordinates of ground arrival point of a wave launched at arbitrary azimuth a and elevation D. In addition, q(d, a) and j(d, a) result from solution of (1) with initial conditions (2), where q 0, j 0 are coordinates of the transmission site. [28] The roots of set (3) are sought as follows: [29] 1. For known coordinates of transmission and reception sites, the trace s geographical azimuth a 0 and range S 0 are found. [30] 2. For a 0 using the bisection method, elevation D 0 is found for which the solution of set (1) gives S(D 0, a 0 ) S 0 l. Here S(D, a) is the range between the transmission site and ground point (D, a). Solution of (1) starts at D = 1.0 with further incrementing by 0.5. [31] 3. For the set SðD; aþ S 0 ¼ 0 yðd; aþ y 0 ¼ 0; ð4þ one step of coordinatewise descent method is performed with initial conditions (D 0, a 0 + da), where y(d, a) is the azimuth of the current ground arrival point. The initial value of da is 15. [32] 4. The values (D 1, a 1 ) are further used as the initial condition for the modified Newton method to find a root of (3) with given error. The accuracy is defined on the condition that the distance between points of arrival being sought and the receiving antenna location does not exceed l/30. [33] 5. The other possibly existing rays are searched for in a similar way by setting da = a [34] 6. Steps 3 5 are iterated while da <15. [35] 7. The procedure of steps 2 through 6 iterates several times as well, with angles D = D 0 until either a ray goes through and out the ionosphere or D [36] The above technique allows us to find practically all roots of set (3), that is, all rays, at acceptable computational expense. [37] Prior to the ray-tracing and boundary value solution, the corrections df o,x are found to the operating frequency [Barabashov and Vertogradov, 2000]. As a result, the operating frequency is replaced by two equivalent ones: f o = f + df o and f x = f df x. The subsequent calculation is then performed for isotropic media. This approach allows finding both magnetoionic components 3of6

4 having frequency f. The correction values are given by approximating polynomial P(S, g, I): df o;x f h ¼ PS; ð g; IÞ ¼ X3 i;j;k¼0 C ðo;xþ i;j;k Si g j I k : Here f h is gyrofrequency, S is path range in 1000 km for ranges below 1000 km and is a unity for longer paths, and geomagnetic azimuth g and magnetic inclination I are given in radians and are brought to the interval [0, p/2]. Coefficients of the polynomial were obtained using results of calculation of the above corrections for paths ranging from 50 to 2000 km for I and g varying within Magnetic parameters f h, I, and g are found in nodes of the space-time grid using components of magnetic field set by the global model. In intermediate points, cubic spline approximation is used Field Strength Calculation [38] Ionospheric absorption L f (f, t) accumulated along the trajectory resulting from solution of the boundary value problem is calculated using a generalized theorem of equivalence [Barabashov and Vertogradov, 1989]. The equivalence relationship between integral absorption values for oblique and vertical propagation in spherical horizontally irregular magnetoactive ionosphere is given by Lðf; n; f h Þ ¼ 0:5 L v1 f v1 ; n 0 1 ; f h1 0 ; h 1 þ L v2 f v2 ; n 0 2 ; f h2 0 ; h 2 : [39] Here L(f, n, f h ) gives the value of oblique absorption at frequency f with collision frequency profile n(h) and gyrofrequency f h, and L v1 (f v1, n 0 1, fh1, 0 h 1 ) and L v2 (f v2, n 0 2, fh2, 0 h 2 ) are values of absorption for vertical incidence into the ionosphere which have been calculated for electronic density distribution profiles at the points of an oblique ray entering and exiting the ionosphere, respectively. Equivalent frequencies f v1,2, collision frequency profiles n 0 1,2(h), and gyro frequencies fh1,2 0 are given by f v1;2 ¼ f sin D 1;2 n 0 1;2 ðhþ ¼ nðhþsin D 1;2 f 0 h1;2 ¼ f h sin D 1;2 : Here elevation angles D 1,2 are taken at the points of a ray crossing the lower bound of the ionosphere, as angles between the geomagnetic field and wave normal. [40] The value of absorption for each normal wave at vertical incidence is calculated by numeric integration of the imaginary part of Appleton s refractive index with respect to altitude using Gaussian 20-point quadrature formulas. Error of calculating collision loss for both magnetoionic components lies within 1 db. [41] Space loss (ray divergence) L S =10lg S 2 e is found by numeric differentiating using a simplified expression of S 2 e : S 2 e ¼ sin D 2 R2 E ðcos a 2 cos D1 ¼const: a1 ¼const Here D 1 and D 2 are launch and arrival angles, R E is the radius of the Earth, and a 1 and a 2 are azimuths at points of a ray entering and exiting the ionosphere. [42] Total multipath field components E n (n =1,2,3)in the local coordinate system at the (x,y) point are E n ¼ X j E j ðt; f ; x; y ÞA jn ðt; f Þexp i2pfp j ðt; f ; x; yþ=c ; ð5þ where j(t, f) is the number of rays (roots of boundary value problem (4)); E j (t, f, x, y) is the amplitude of the jth ray as a function of time t, frequency f, and coordinates x, y; A jn (t, f) are coefficients of conversion of the physical to local coordinate system; and P j (t, f, x, y) is the phase delay of the jth ray. [43] Amplitude of the jth ray is given (with respect to 1 mv/m), by E j ¼ 107:8 þ 10 lg P L L s L p L r þ K 1 ; where P is transmit power in kw, L is ionospheric absorption in db, L s is space loss in db, L r is Earth reflection loss in db, and L p is loss in db due to mismatch between polarization ellipses of a wave coming to the ionosphere and a magnetoionic component excited in the ionosphere. The latter loss is sought within the theory of the limiting polarization [Davies, 1990]. The values of E j (t, f, x, y) are converted to mv/m for determination of E n. [44] The variation of total field E(t) with time is obtained using series of E j (t) and P j (t, f, x, y) calculated with time increment of 0.1 s. This value of time sampling is selected because the maximum Doppler offset of operating frequency does not exceed 1 Hz. Then the E(t) series is obtained using (5). Validity of basic points of the model and properties of wave field predicted using the model are confirmed by a variety of experimental data available at present. 3. Test Results and Conclusions [45] A few figures illustrate the results of simulating space-time distribution of wave field on the Earth s surface for the model with two TIDs, typical for middle latitudes: d = 10%, L = 200 km, T = 20 min, b = 45, 4of6

5 Figure 2. Field strength time variation. [47] Figures 1a and 1b depict simulated equal phase curves of total multipath field in 2p spacing on a m area. Curves 1, 2, and 3 correspond to successive time increments of 0.1 s. Figure 2 shows simulated time variation of field strength in a local reception point for a period of 300 s, with radiated power of 1 kw. Figure 3 shows the autocorrelation ratio of field strength at the same location. [48] The present model can suggest various areas of application. It can be used in the following situations: to obtain complete information on mode, ray, polarization, and space-time structure of the field for specific heliophysical and geophysical conditions; to compute the time series of complex voltage on every element of an antenna array of any size and configuration, which enables us to study spectral, statistical, and correlative (spatial and time) properties of ionosphere-reflected signals; to study both narrowband and wideband signal propagation via the ionospheric channel; to set up requirements for antenna arrays in communication and direction finding; to make expert estimates of algorithms for space-time processing of signals in antenna arrays; to calculate amplitude frequency and phase frequency channel responses and their variation with time; and to run signals through an ionospheric channel. Figure 1. Equal phase curves: (a) vertical plane and (b) ground plane. g =45, and F =0 ; and d = 5%, L = 100 km, T = 30 min, b = 30, g =0, and F =90. [46] The modeling was performed for the following conditions: path range is 1000 km; arc of the great circle connecting the transmission and reception sites is parallel to the x axis; the operating frequency is 9.0 MHz; for the case of 1F2 and 2F2, rays are simultaneously present with close amplitudes; and elevation angles are 23 and 53, respectively. 5of6 Figure 3. Simulated autocorrelation ratio.

6 [49] Acknowledgment. The authors thank J. M. Goodman (Radio Propagation Services, Inc.) for helpful remarks. References Barabashov, B. G., and G. G. Vertogradov (1989), Generalization of equivalence theorem of skywave absorption for spherical magneto-active ionosphere (in Russian), Proc. Inst. Eng. Radiocommun., 2, Barabashov, B. G., and G. G. Vertogradov (1996), Dynamical adaptive physically-structural model of ionospheric skywave channel (in Russian), Math. Model., 8(3), Barabashov, B. G., and G. G. Vertogradov (2000), Structural physical model of ionospheric channel, paper presented at Millennium Conference on Antennas and Propagation, Eur. Space Agency, Davos, Switzerland, 9 14 April. Davies, K. (1990), Ionospheric Radio, Peter Peregrinus, London. Goodman, J. M. (1992), HF Communication: Science and Technology, Van Nostrand Reinhold, Hoboken, N. J. M. M. Anishin, B. G. Barabashov, and O. Y. Pelevin, Department of Radio Physics, Faculty of Physics, Rostov State University, Rostov-on-Don , Russia. (ope@jeo.ru) 6of6