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.)  0 and  0 are the electrical permittivity and magnetic inductive capacity of free space, and are wavelength & frequency The atmosphere modifies EM propagation:  1, is larger than  0, and furthermore,  1 is a function of height. Speed of propagation is The waves bend in the atmosphere (the process of refraction) due to the variation of  1 (x i ) =  1 (x,y,z). Note  atmos  1 (>  0 ), 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 O 2 and N 2 are not polarized, but they can become polarized in the presence of an imposed electric field. For no external forces, the orientation of H 2 O molecules is random due to thermal agitation. However, if H 2 O subjected to an E field, then it is aligned so that the dipole fields add constructively to enhance the net electric force on each H 2 O 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, N , multiplied by a factor  T proportional to the molecule ’ s level of polarization, expressed by the Lorenz-Lorentz formula  r is the relative permittivity,  r =  /  0 = n 2. For air, the value of  r 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  is related to the number density N  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 (3 rd ) 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): c d = 77.6 K mb -1, c w1 = 71.6 K mb -1, c w2 = 3.7 x 10 5 K 2 mb -1 (2.17) can be approximated as Example: e = 10 mb, p = 1000 mb, T = 300 K N = 0.26(10 3 + 1.6 x 10 2 ) = 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  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): a)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: a)n is linearly dependent on h b)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, H 0 - 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.