EM propagation paths

1. EM propagation paths 1/17/12

2. Introduction • Motivation: For all remote sensing instruments, an understanding of propagation is necessary to properly interpret the measurements. • EM waves propagate as straight lines at the speed of light (c) (recall Maxwell’s Eqs.) • e0 and m0 are the electrical permittivity and magnetic inductive capacity of free space,  and  are wavelength & frequency • The atmosphere modifies EM propagation: e1, is larger than e0, and furthermore, e1 is a function of height. Speed of propagation is • The waves bend in the atmosphere (the process of refraction) due to the variation of e1(xi) = e1(x,y,z). • Note matmos  1 (>m0), and therefore v < c

3. Refractive index of air • Index of refraction, n, is defined as the ratio of the speed of light in a medium to that in a vacuum: • n is is dependent on the density and polarization of molecules • O2 and N2 are not polarized, but they can become polarized in the presence of an imposed electric field. • For no external forces, the orientation of H2O molecules is random due to thermal agitation. However, if H2O subjected to an E field, then it is aligned so that the dipole fields add constructively to enhance the net electric force on each H2O molecule. • This behavior is related to the extraction of energy from the incident wave and leads to attenuation (e.g., 95 GHz)

4. permittivity of a gas depends on molecular number density, Nr, multiplied by a factor aT proportional to the molecule’s level of polarization, expressed by the Lorenz-Lorentz formula • er is the relative permittivity, er = e/e0 = n2. • For air, the value of er is 1.000300, and the above formula can be rewritten as • For the atmosphere which is a mixture of molecules, the following equation applies: • mass density r is related to the number density Nr by the molecular weight M: • Normalized equation of state for a gas, for standard temperature (273 K) and pressure (1013.25 mb)

5. From Avogadro’s Law, the number of molecules per unit volume of gas is given as • and the number for an arbitrary T and p is • The last (3rd) term on the rhs represents the contribution from the permanent dipole moment of water vapor (3.3)

6. Refractivity • Define: • From the above, n becomes • Expansion in a Taylor’s series: • Using (3.3): • cd = 77.6 K mb-1, cw1 = 71.6 K mb-1, cw2 = 3.7 x 105 K2 mb-1 • (2.17) can be approximated as • Example: e = 10 mb, p = 1000 mb, T = 300 K N = 0.26(103 + 1.6x102) = 300 n = 1 + Nx10-6 = 1.000300 • The value of n in the atmosphere differs little from that of free space, but this small difference, and the variation with height, is important to EM propagation. (2.17)

7. N normally decreases with altitude, since both p and T decrease with altitude, on the average, and p dereases at a more rapid rate (i.e., the fractional decrease is much larger). When dN/dz < -157 km-1 (the case for inversion layers), EM rays are bent toward the earth’s surface. Small scale fluctuations in N  Bragg scattering (discussed later). We will first consider quasi-horizontal layers of N and how dN/dz affects EM propagation in the atmosphere.

8. Spherically-stratified atmosphere • Assume T and e (RH) are horizontally homogeneous so that N = N(z). • A ray path in spherically stratified atmosphere is given by the differential equation • Whose solution is

9. The variable s(z) is the great circle distance to a point directly below the ray at height h above the surface, a is the earth’s radius, R is the radial distance from the center of the earth, and a is elevation angle of the radar antenna. We also assume: n(z) is smoothly changing so that ray theory is applicable n(0) is the refractivity at the radar site

10. Aside: Snell’s Law • The variation of a ray path with a change in refractive index is given by In the atmosphere, n decreases with height, and therefore the rays are bent toward the earth (as in the figure above).

11. Equivalent earth model • Several simplifications can be applied to (3.5): • Small angle approximation: << 1 • b) Large earth approximation: z << R • Then the approximate equations describing the path of ray at small angles relative to the earth are:

12. The index of refraction for the standard atmosphere is dn/dz = const = -4 x 10-8 m-1. For the standard atmosphere, one can define a fictitious earth curvature where rays propagating relative to the fictitious earth are straight lines, as follows:

13. The height of a ray as a function of slant range for zero elevation angle is given by (3.7) • This relation assumes that: • n is linearly dependent on h • The development of Eq. (3.7) assumed dz/ds <<1, which imposes a limit on the use of an effective earth radius

14. The vertical gradient of n is typically not constant, and appreciable departures from linearity exist in the vicinity of temperature inversions and large vertical gradients in water vapor. The departure between the 4/3 earth radius model and a reference atmosphere is shown in Fig. 2.7 below. In each model, the surface value is N = 313. A large difference between these two models exists at heights > 2 km AGL.

15. For weather radar applications (z < 10 km), and n exhibits a vertical gradient of -1/4a within the lowest one kilometer, the 4/3 earth radius model can be used with sufficient accuracy. Fig. 2.8 reveals a comparison of ray paths for two models: the 4a/3 model, and an exponential model of the form The 4a/3 model works well, except in low level inversions

16. Standard refraction Rinehart provides the following equation to compute beam height when standard refraction applies: r - slant range ,  - elevation angle, H0 - height of the radar antenna R’ = (4/3)R, R - earth’s radius (6374 km)

17. Ground-based ducts and reflection heights The example profile of N shown in Fig. 2.9 illustrates anomalous propagation and beam distortions. Ray paths are shown in Fig. 2.10

18. Example of anomalous propagation

19. Homework • Do problems 1-3 in the notes.