Improved Fourier transform methods for solving the parabolic wave equation

RADIO SCIECE, VOL. 37, O., 11, 1.19/1RS488, Improved Fourier transform methods for solving the parabolic wave equation James R. Kuttler Johns Hopkins Applied Physics Laboratory, Laurel, Maryland, USA Ramakrishna Janaswamy 1 aval Postgraduate School, Monterey, California, USA Received May 1; revised September 1; accepted September 1; published 1 March. [1] A standard method for modeling electromagnetic propagation in the troposphere is the Fourier split-step algorithm for solving the parabolic wave equation. An important advance in this technique was the introduction of the mixed Fourier transform, which permitted the extension of the method from propagation over only smooth perfectly conducting surfaces to quite general surfaces with impedance boundary conditions. This paper describes improvements in the implementation of the mixed Fourier transform, which make the method more robust and efficient and avoid potential numerical instabilities, which occasionally caused problems in the previous implementation. Some examples are also presented. IDEX TERMS 644 Electromagnetics umerical methods; 689 Electromagnetics Wave propagation (475); 694 Radio Science Atmospheric propagation; 6964 Radio Science Radio wave propagation 1. Introduction [] The parabolic approximation/fourier split-step algorithm [Ko et al., 1983; Dockery, 1988] is a powerful method for modeling electromagnetic propagation through inhomogeneous atmosphere and above the surface of the Earth, which may have large- and small-scale roughness plus various dielectric properties. Assuming azimuthal symmetry, Maxwell s equations can be reduced to a two-dimensional scalar Helmholtz equation for the transverse component of either the electric field or the magnetic field, depending on whether the polarization is horizontal or vertical, respectively. The Helmholtz equation is then factored into forward and backward propagating pieces, and only the forward propagating part is used. This is a parabolic wave equation, which is solved by marching over range steps. At each step the field component at range x + Dx is obtained from the field component at range x by the action of an operator which is split into a product of three operators (hence the name split-step ), uxþ ð DxÞ ¼ ABAuðÞ x 1 ow at University of Massachusetts, Amherst, Massachusetts, USA. Copyright by the American Geophysical Union. 48-664//1RS488 5-1 The symmetrically split out A operators account for all of the atmospheric variation and large-scale surface roughness, while the central B operator propagates the field as though in a vacuum above a horizontal impedance plane [Kuttler, 1999] and is the object of interest here. [3] The action of the B operator is implemented by Fourier transform methods. An important advance in this technique was the introduction of the mixed Fourier transform [Kuttler and Dockery, 1991]. This has permitted the extension of the split-step method from propagation over only smooth perfectly conducting surfaces to surfaces satisfying impedance boundary conditions. The split-step approach has also been extended to general terrain problems [Donohue and Kuttler, ]. [4] The algorithms used to implement the mixed Fourier transform have been improved in the years since its introduction to make it faster and more robust [Dockery and Kuttler, 1996]. However, certain combinations of parameters can occasionally cause numerical instabilities in the method. This paper will give some examples of the occurence of these instabilities and present some new algorithms for implementing the mixed Fourier transform, which will overcome these numerical problems and also speed up the method. These algorithms will be referred to as the forward difference and the backward difference methods for the

5 - KUTTLER AD JAASWAMY IMPROVED FOURIER TRASFORM METHODS mixed Fourier transform, and they are most effective when employed in complement to each other.. Mixed Fourier Transform (MFT) [5] The function u(z), which is to be decomposed by Fourier transform methods, is defined on z < 1, obeys the requirement that u(z)! asz!1, and satisfies the impedance boundary condition du þ a u ¼ z ¼ ; ð1þ where a is a given complex constant. Introduce the auxillary function wz ðþ¼ duðþ z þ a uz ðþ z 1 ðþ ote that w(z)! asz!1also and w vanishes at z =. ow consider the Fourier sine transform of w, UðpÞ ¼ wz ðþsin pz ð3þ Putting () into (3) and integrating by parts, it can be seen that UðpÞ ¼ uz ðþa ½ sin pz p cos pzš; which is called the mixed Fourier transform (MFT) of u for obvious reasons. [6] To invert this transform, first invert the sine transform (3) to get wz ðþ¼ p UðpÞ sin pz dp ext recover u from w by solving the differential equation (). This leads to uz ðþ¼ p a sin pz p cos pz UðpÞ a þ p dp ð4þ This is the inverse of the MFT when Re (a) <. However, when Re (a) >, there is a homogeneous solution Ke -a z of () which must be added to (4). Then the inverse of the MFT for Re (a) >is uðzþ ¼K e az þ p Uð pþ a sin pz p cos pz a þ p dp ð5þ To determine the constant K, multiply both sides of (5) by e a z and integrate over z < 1 to see that K ¼ a where the orthogonality relation uz ðþe a z ; e az ½a sin pz p cos pzš ¼ ð6þ was used. In radio frequency propagation, generally, Re (a) > for vertical polarization (v-pol) and Re (a)< for horizontal polarization (h-pol). 3. Discrete Mixed Fourier Transform (DMFT) [7] The earliest method used to numerically implement the MFT sampled the function u(z) at points n, n =, and used discrete sine and cosine transforms to give the approximation UðjdpÞ ¼ X j ¼ ; uðn Þ a sin p jn p jn jdp cos where dp = p/ and the prime on the sum means that the first and last values are to be weighted with factor 1/. The inverse is uðnþ ¼ Ke a n þ X uðjþ j¼ n ¼ ; where K ¼ a X p jn a sin jdp cos p jn a þ ðjdpþ e an uðnþ [8] An attempt at introducing an auxiliary function w(n) satisfying an analog of (), so that only a sine transform needs to be used in the algorithm, was unsuccessful because the solution of the difference equation was numerically unstable. It was also eventually discovered that the naive implementation of the DMFT described above could be numerically unstable if Re (a) became small, so that the term a + p for p Im (a). The cause of this instability was believed to be the loss of an orthogonality condition analogous to (6) between e an and the discrete functions

KUTTLER AD JAASWAMY IMPROVED FOURIER TRASFORM METHODS 5-3 a sin p jn p jn jdp cos [9] To restore this orthogonality, the following modification was made to the DMFT " # UðjdpÞ ¼ X uðnþ a sin p jn pj sin cos p jn j ¼ ; with inverse uðnþ ¼ X j¼ uðjþ a sin p jn 1 sin p j a þ 1 sin p j þ C 1 r n þ C ð rþ n n ¼ ; where r is the root of r þ ar 1 ¼ p jn cos of smaller magnitude (the other root being r 1 ). Then r n and ( r) n are both orthogonal to a sin p jn 1 sin p j p jn cos with respect to X j ¼ 1 1; ð7þ [1] As an additional bonus to the improved DMFT, an analog of () can now be used by defining wðnþ ¼ u ½ ðnþ 1Þ Š u ½ð n 1 ÞŠ n ¼ 1 1; with w() = w() =. Then UðjdpÞ ¼ X wðnþ sin p jn þ a uðnþ j ¼ ð8þ To recover u from the auxillary function w, solve (8). This is now an exact relation, not an approximation, so no numerical problems are caused. Although this is a difference analog of a first-order differential equation, it is a second-order difference equation. Thus the general solution consists of a particular solution plus a linear combination of two independent solutions of the homogeneous equation. These two solutions are r n and ( r) n. Approximating a first-order differential equation by a secord-order difference equation can lead to numerical instability. A disussion of this phenomenon is given, e.g., by Hildebrand [1987, section 6.7]. 4. Bad Alpha Problem [11] Although much more stable than the earlier version of the DMFT and with the advantage of only requiring a sine transform, the above version still exhibits occasional numerical instability when Re (a)!, which is generally characterized by jrj!1. This has come to be known as the bad alpha problem. It occurs sometimes when using the Miller-Brown formula [Miller et al., 1984] for modeling a rough surface by adjusting the Fresnel reflection coefficient, thereby changing the effective impedance of the surface and making a a rangedependent variable. [1] A spectacular example of the bad alpha problem is shown in Figure 1, in which the above version of the DMFT was used in a split-step method for a 1-GHz, h- pol problem. The maximum problem altitude is feet (61 m), and an aperture antenna producing a sinc beam is located at 1 feet (3 m) altitude and pointed horizontally. This is in free space over a flat Earth. The transform size is 51 and Miller-Brown was used for an RMS wave height of 5 m (16.4 feet). It can be seen why sometimes the bad alpha problem is also called the big V problem. The correct propagation factor for this problem, computed by the method to be discussed, is shown in Figure. 5. Forward Difference DMFT [13] To obtain a different formulation of the DMFT, replace the central difference equation (8) with a forward difference equation wðnþ ¼ u½ðnþ 1Þ Š u ð n Þ n ¼ 1 1; þ a uðnþ with w() = w() =. This has the advantage of being only a first-order difference equation, thus requiring only one homogeneous solution in the general solution. To analyze the forward difference algorithm,

5-4 KUTTLER AD JAASWAMY IMPROVED FOURIER TRASFORM METHODS Figure 1. The big V is an occurrence of the bad alpha problem for 1-GHz h-pol, 5-m RMS roughness. define r =1 a. Then, multiply the above equation by and redefine w by wðnþ ¼ u½ðnþ 1ÞŠ r uðnþ n ¼ 1 1 ð9þ The solution to the homogeneous equation is r n (note that r has a different meaning here than in the previous DMFT algorithm). ow let Uðjdp Þ ¼ X 1 ¼ X 1 ¼ X wðnþsin p jn fu½ðnþ 1ÞŠ ruðnþgsin p jn uðnþ sin p jn ð 1Þ r sin p jn This then is the forward difference DMFT. To see a connection to the previous method, write sin p jn ð 1Þ r sin p jn ¼ cos p j r sin p jn sin p j use r =1 a, and divide by to get " # a sin p j cos p jn ; sin p jn 1 sin p j p jn cos ow compare these functions with the ones in (7). It is easy to see the orthogonality of these functions with r n, X r n sin p jn ð 1Þ r sin p jn ¼

KUTTLER AD JAASWAMY IMPROVED FOURIER TRASFORM METHODS 5-5 Figure. Correct propagation for the problem of Figure 1 is shown. Moreover, these functions can be shown to be mutually orthogonal, where here orthogonality is with respect to X [14] To completely specify the forward difference DMFT requires computing the quantity C ¼ X r n uðnþ ð1þ To invert the forward difference DMFT, the procedure is reversed. First, inverse sine transform U to get w. Then, get a particular solution u p of the first-order difference equation (9) by setting u p () = and using u p ½ðn þ 1ÞŠ ¼ wðnþþru p ðnþ n ¼ 1 ð11þ Then u = u p + Ar n, where A is determined by requiring that (1) be satisfied. [15] The forward difference algorithm works best when jrj < 1. Since jj r ¼ ½1 ReðaÞŠ þ½imðaþš ; this condition is equivalent to Re ðaþ jaj ote that this can never hold for h-pol and even for v-pol could impose a severe restriction on the size of (and hence, the transform size) if Re (a)!, as can happen when using Miller-Brown. [16] ow it is still possible to use the forward difference algorithm when jrj > 1. In that case it is prudent to

5-6 KUTTLER AD JAASWAMY IMPROVED FOURIER TRASFORM METHODS write the homogeneous solution as (1/r) -n, and replace (1) with C ¼ X 1 1 n u½ð nþš ð1þ r Instead of (11), get a particular solution u p by setting u p () = and use u p ½ð nþš ¼ 1 u p f½ ðn 1ÞŠg r w½ð nþšg n ¼ 1 Then u = u p + A(1/r) n, where Ais determined by requiring that (1) be satisfied. Even with these modifications, the forward difference algorithm does not perform well when jrj gets too large, as will be seen. 6. Backward Difference DMFT [17] After analyzing the forward difference algorithm, a natural idea is to try a backward difference formulation. Instead of (8), consider wðnþ ¼ u ð n Þ u ½ð n 1 ÞŠ n ¼ 1 1; þ auðnþ ð13þ with w() = w() =. Again, this is only a first-order difference equation. ow we define r =(1+a) 1, rearrange (13), and redefine w by wðnþ ¼ uðnþ ru½ðn 1ÞŠ n ¼ 1 1 ð14þ Again the solution to the homogeneous equation is r n, where r again has a different meaning here than in the previous two algorithms. Let Uðjdp Þ ¼ X 1 ¼ X 1 ¼ X 1 wðnþsin p jn fuðnþ ru½ðn 1ÞŠgsin p jn uðn Þ sin p jn p jnþ1 r sin ð Þ This is the backward difference DMFT. Again the functions sin p jn p jnþ r sin ð 1 Þ are easily seen to be orthogonal to r n and mutually orthogonal, where here orthogonality is with respect to X 1 [18] To completely specify the forward difference DMFT requires computing the quantity C ¼ X 1 r n un ð Þ ð15þ Inverting the backward difference DMFT uses steps similar to those of the forward difference algorithm First, inverse sine transform U to get w. Then get a particular solution u p of the first-order difference equation (14) by setting u p () = and using u p ðnþ ¼ wðnþþru p ½ðn 1ÞŠ n ¼ 1 Then u = u p + Ar n, where A is determined by requiring that (15) be satisfied. [19] Again, the backward difference algorithm works best when jrj < 1. Since r 1 ¼ ½ 1 þ Re ð a ÞŠ þ½imðaþš ; this condition is equivalent to > ReðaÞ jaj This is always true for v-pol (when Re (a) > ) and imposes a lower bound on the size of for h-pol (when Re (a) < ). Again, it is still possible to use the backward difference algorithm when jrj > 1 with the same sort of modifications as were made in the forward difference case. In general, the backward difference algorithm can be used without numerical problems in many more cases than can the forward difference algorithm. 7. Central Difference DMFT [] One possible drawback to the forward and backward difference algorithms is that they only approximate the impedance boundary condition (1) to first order in, whereas in the old version of the DMFT it is approximated to second order in. A natural question is whether there is a first-order difference equation that approximates (1) to second order in? The answer is yes, provided calculations are performed on a mesh of points that is shifted by / from the usual mesh. That

KUTTLER AD JAASWAMY IMPROVED FOURIER TRASFORM METHODS 5-7 is, sample u(z) at the points (n + 1/), n = 1, and define wðnþ ¼ u nþ 1 u n 1 þ a u nþ 1 þ u n 1 ð16þ n ¼ 1 1; with w() = w() =. ow define r =(1 a/)/(1 + a/), rearrange (16), and redefine w by wðnþ ¼ u nþ 1 ru n 1 n ¼ 1 1 ð17þ Again, the solution to the homogeneous equation is r n, where r has yet a different meaning here than in the previous three algorithms. Let Uðjdp Þ ¼ X 1 ¼ X 1 ¼ X 1 wðnþsin p jn u nþ 1 ru n 1 sin p jn u nþ 1 sin p jn p jnþ r sin ð 1 Þ So this is the central difference DMFT. Again the functions sin p jn p jnþ r sin ð 1 Þ are easily seen to be orthogonal to r n and mutually orthogonal, where here orthogonality is with respect to X 1 [1] To completely specify the central difference DMFT requires computing the quantity C ¼ X 1 r n u nþ 1 ð18þ Inverting the central difference DMFT uses steps similar to those of the forward and backward difference algorithms. First, inverse sine transform U to get w. Then get a particular solution u p of the first-order difference equation (17) by setting u p (/) = and using u p n þ 1 ¼ wðnþþru p n 1 n ¼ 1 Then u = u p + Ar n, where A is determined by requiring that (18) be satisfied. [] There is one additional step needed in the central difference algorithm to produce a value for uon the surface. This value is not needed in the computation but is generally a quantity desired in the output. Thus (17) is used at n =, with left side zero, to get a value for u at the virtual point n/, and then uðþ ¼ 1 ½u= ð Þþu ð = ÞŠ The result is uðþ ¼ u= ð Þ 1 1 a [3] Again, the central difference algorithm works best when jrj < 1. Since jj r ¼ h 1 þ 1 i þ h i ReðaÞ 1 ImðaÞ h 1 1 i þ h i ReðaÞ 1 ; ImðaÞ this condition is equivalent to Re (a) >, which is always true for v-pol. Again, it is still possible to use the central difference algorithm when jrj > 1 with the same sort of modifications as were made in the forward difference case. The central difference algorithm can also be used without numerical problems in many more cases without numerical problems than can the forward difference algorithm. 8. Bad Alpha Regions [4] How big can jrj be before these algorithms cease to work? Apparently, the criterion is when jr j exceeds the smallest number that causes overflow in the language and platform on which the computation is done. For example, for MATLAB on a PC, this number is 14. Thus the limit on jrj in this case is 14/, which, for a transform size = 56, is 16, but, for a transform size = 15, is only 1.19. Remember that the r in all the algorithms, including the older version of the DMFT,

5-8 KUTTLER AD JAASWAMY IMPROVED FOURIER TRASFORM METHODS Figure 3. Shaded areas are bad alpha regions for d max = 1 and feet (35 and 61 m). depend on, which, for a fixed height of the calculation region, is inversely proportional to. [5] Letting R denote this maximum value of jrj, the meaning of jrj < R, using the different definitions of r for the algorithms above, is as follows. For the forward difference algorithm, ReðaÞ 1 þ½imðaþš < R ; ð19þ which is the interior of a circle of radius R/ centered at 1/ in the right half of the complex a plane. For the backward difference algorithm, ReðaÞþ 1 þ½imðaþš > 1 ; ðþ R which is the exterior of a circle of radius 1/R centered at 1/ in the left half of the complex a plane. Finally, for the central difference algorithm, ReðaÞþ R þ 1 R 1 þ Im a ½ ð ÞŠ > 4 R R ; 1 which is also the exterior of a circle with center in the left half plane. [6] In general, the good region for the backward difference is larger than the good region for the central difference, as can be seen by comparing the radii of their circles. Furthermore, the good regions for both of these algorithms are larger than the good region for the forward difference, since the latter is the interior of a circle, rather than the exterior. However, none of these regions completely contains another one. That is, there are always values of a that will be inside any one of the

KUTTLER AD JAASWAMY IMPROVED FOURIER TRASFORM METHODS 5-9 Figure 4. Forward difference is going bad for -GHz h-pol, 3-m RMS roughness. good regions but outside all of the other good regions. To see this, consider just the left-most point of the forward difference circle and the right-most points of the central and backward difference circles, all on the Re (a) axis. These are 1 R < 1 R 1 þ R < 1 R R ; respectively. Since each algorithm s good region lies to the right of these points, respectively, the assertion is verified. [7] Figure 3 plots some of these good and bad regions in the complex alpha plane for the case = 14. First, for a region with maximum height d max = feet (61 m) ( = d max /), the largest circle in Figure 3 bounds the good region for the forward difference algorithm (good is inside). The small circle that it intersects bounds the good region for the backward difference algorithm (good is outside). Thus, by using either the backward or the forward difference algorithms as appropriate, the combined bad alpha region, which is the intersection of the bad alpha regions of each, is the shaded lune inside the small circle and outside the large circle. In the same figure is the case in which d max is reduced to 1 feet (35 m) (thus halving ). otice that the combined bad alpha region in this case is disjoint from the previous one. Thus a bad alpha can always be avoided by appropriate choice of backward or forward difference algorithms and computational domain height d max. 9. Concluding Discussion [8] In a series of experiments with parabolic wave equation (PWE) routines using the forward, backward, and central difference algorithms, it was generally observed that when the central difference algorithm

5-1 KUTTLER AD JAASWAMY IMPROVED FOURIER TRASFORM METHODS Figure 5. Backward difference gives correct results for the problem of Figure 4. worked, so did the backward difference algorithm and they gave virtually identical results. Thus, despite the fact that theoretically the central difference algorithm better approximates the impedance boundary condition, we are not currently using the central difference algorithm, because it requires calculating on a shifted grid, causing some bookkeeping and input/output nuisance. (Of course, further testing may produce cases in which the more accurate boundary condition is required.) [9] As expected, there are cases in which the backward difference algorithm gives the right answer and the forward difference algorithm does not. In Figure 4 there is a -GHz, h-pol antenna at 1 feet (3 m), d max = 5 feet (15 m), = 48, and the RMS wave height is 3 m (9.84 feet). The forward difference algorithm is clearly going bad. However, in Figure 5 the backward difference algorithm gets the correct result. (The PWE routine using the older version of the DMFT cannot do this problem at all. It blows up almost immediately.) [3] However, as shown above and in Figure 3, there are potentially cases when we can be in the bad region for the backward difference algorithm but in the good region for the forward difference algorithm. Therefore the recommended method for fixing the bad alpha problem, based on the above observations, is to determine R (which depends on the computing platform) and check to see if () is satisfied. (This need only be checked for h-pol, as it will always be true for v-pol.) If it is, use the backward difference algorithm. If not, see if (19) might be satisfied. If it is, then use the forward difference algorithm. These are both defined on the usual (mdx, n) mesh, so there is no difficulty in switching between them. If (19) is also not satisfied, change the transform size. [31] Extensive testing performed so far on this backward plus forward difference algorithm has shown the

KUTTLER AD JAASWAMY IMPROVED FOURIER TRASFORM METHODS 5-11 method to be extremely robust. Additional testing will still be done, but it appears that this is a major advance in solving the bad alpha problem. Also, not only is this new algorithm more robust than the older version, but it will be slightly faster as well, because only a single backsolve and one homogeneous solution calculation are needed per step, rather than two for the older version. References Dockery, G. D., Modeling electromagnetic wave propagation in the troposphere using the parabolic equation, IEEE Trans. Antennas Propag., 36(1), 1464 147, 1988. Dockery, G. D., and J. R. Kuttler, An improved impedanceboundary algorithm for Fourier split-step solutions of the parabolic wave equation, IEEE Trans. Antennas Propag., 44(1), 159 1599, 1996. Donohue, D. J., and J. R. Kuttler, Propagation modeling over terrain using the parabolic wave equation, IEEE Trans. Antennas Propag., 48(), 177 6,. Hildebrand, F. B., Introduction to umerical Analysis, nd ed., Dover, Mineola,. Y., 1987. Ko, H. W., J. W. Sari, and J. P. Skura, Anomalous microwave propagation through atmospheric ducts, Johns Hopkins APL Tech. Dig., 4(), 1 6, 1983. Kuttler, J. R., Differences between the narrow-angle and wideangle propagators in the split-step Fourier solution of the parabolic wave equation, IEEE Trans. Antennas Propag., 47(7), 1131 114, 1999. Kuttler, J. R., and G. D. Dockery, Theoretical description of the parabolic approximation/fourier split-step method of representing electromagnetic propagation in the troposphere, Radio Sci., 6(), 381 393, 1991. Miller, A. R., R. M. Brown, and E. Vegh, ew derivation for the rough-surface reflection coefficient and for the distribution of sea-wave elevations, IEE Proc., Part H, Microwaves Opt. Antennas, 131(), 114 116, 1984. R. Janaswamy, Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, 15 A Marcus Hall, Amherst, MA 13, USA. (janaswamy@ecs. umass.edu) J. R. Kuttler, Applied Physics Laboratory, 111 Johns Hopkins Road, Laurel, MD 73-699, USA. ( jim.kuttler@ jhuapl.edu)