MET 358 – Radar Meteorology

1. MET 358 – Radar Meteorology EM Theory – Polarization and Refraction Lecture 3

2. Polarization • In Lecture 2, we showed that the properties of an EM wave can be described by either the electric or magnetic fields alone • Tend to prefer electric, since water molecules have strong electric dipole moments (O-H bonds) • Most backscattered energy from water comes from vibrations of O-H bonds caused by the passing electric field • This convention will be used in subsequent work

3. Polarization • Polarization refers specifically to the orientation of the EM wave – which can be described using either electric or magnetic field vector • Doviak and Zrnic (1993) • Plane of polarization – plane containing electric field • Let’s assume an EM wave propagates on the z-axis, and the electric field is found on the x-y plane

4. Polarization Here, we see three different orientations, as shown by Doviak and Zrnic (1993, p. 12). In (a), the electric field is oriented such that the oscillation is entirely on the y-axis. In (b), the electric field is oriented such that the oscillation is entirely on the x-axis. In (c), we see the wave propagates with a circular pattern in the x-y plane. In the next slide, the mathematics behind these configurations will be shown.

5. Polarization • Consider an electric field with components Ex and Ey on the x- and y-axes, respectively. Using the standard symbols for linear frequency and time, and using a phase difference δ between the x- and y-axes:

6. Polarization • So, referring to the figure, in (a), we assume Exm=0. • In (b), we assume Eym=0. • In (c), we assume Exm=Eym, and δ = ±π/2. (Specifically, for right-hand circular, δ=π/2.)

7. Polarization • If δ = 0 or π, the oscillation will be back and forth on an axis determined by the relative values of Exm and Eym. • For anything else, the oscillation is elliptical. • Most radars are horizontally polarized.

8. Polarization • However, many research radars have the ability to vary polarization – important in diagnosing characteristics of weather targets. • Dual-polarization, e.g., based on varying the orientation of electric fields (horizontally and vertically). More on this later in the class.

9. Refraction • In Lecture 2, we “derived” the propagation speed for electromagnetic waves in a vacuum: • For any medium, the velocity of the EM wave is:

10. Refraction • The velocity v is lower in a medium other than a vacuum because of the permittivity and permeability properties of that medium. • We define the ratio of an EM wave in a vacuum to that in the medium of interest as index of refraction:

11. Refraction • For air at sea level, n=1.0003 • Total refraction also involves absorption, which is also based on the properties of the medium: • where k is based on the absorptive properties of the medium • Important when we discuss attenuation (later) • For now, just talking about n

12. Refraction • n depends on the density and polarization of molecules • For density, more dense materials have higher indices of refraction • As such, n decreases with height (~1.0003 near surface to ~1.00 near top of atmosphere) • EM wavefronts bend toward Earth’s surface

13. Refraction • Polarization (in this case) refers to a molecule’s ability to create their own electric field in the absence of external forces • Water (vapor) is one of those molecules • Thus, n is dependent on p and T (density dependence – ideal gas law), and e (vapor pressure – polarization)

14. Refraction • Empirically derived formula for n: • K1 = 7.76*10-5 K/mb • K2 = 5.6*10-6 K/mb • K3 = 0.375 K2/mb • n can be expressed as radio refractivity N • To convert equation for n to N, subtract by 1 and take answer times 106 • i.e., N=(n-1)*106

15. Refraction • If we assume the speed of an EM wave in the Earth’s atmosphere is of a wave in a vacuum, we will accumulate error in the location of a target as the pulse goes farther from the radar • Using n=1.0003, using c=3.0 * 108 m s-1 as an estimation of the wave speed, at a distance of 150 km from the radar, we would accumulate about 45 m of error (0.03%). Not a big problem for local forecasters (about half a football field). • The much MUCH bigger problem is with the height of the radar pulse as it extends farther from the radar

16. Refraction • Hartree et al. (1946) derived the path of a ray in a spherically stratified atmosphere: where h is the height of the beam above Earth’s surface, s is the distance along Earth’s surface, RE is Earth’s radius dh/ds is simply the tangent of the angle of the beam:

17. Refraction • Need to make some assumptions to make this equation usable. • First, we’ll assume that RE >> h, so that higher order terms with their sum can be reduced to RE. • Second, assume that φ is small. Using the Taylor series expansion (HW 1), this means that:

18. Refraction • Third, n ≈ 1 • Using these three assumptions, the above equation reduces to  • M is the modified index of refraction (M = [(h/R)+(n-1)]*106)

19. Refraction • But our equations feature two unknowns (Φ1 and h1) – fortunately can relate Φto h using approximation of Φ equaling dh/ds, as shown earlier • Now let’s look at specific examples for atmospheric conditions…

20. Refraction • First, consider no atmosphere…that is, dn/dh = 0 everywhere • Notice how we can substitute dh/ds in the analysis. • Remember, the initial height is not necessarily zero. Most radar antennas are not at ground level! • Only works with no atmosphere. In other words, the answer typically is not this easy. • Also, what is s exactly? Not the same thing as slant range (r).

21. Refraction • Now, we will consider a “standard atmosphere”, which is one in which dn/dh = -4*10-8 m-1. We also define R’ such that:

22. Refraction • Problem: We still don’t know what s is! Need to relate it to slant range (straight distance a radar beam would traverse over a given time), or relate h to slant range without requiring knowledge of s… • Use trigonometry (and a chalk board)! • Requires a risky assumption…the slant range is approximately straight. • Also assume ho << R’  note we add ho in the last step, however

23. Refraction • Problem: We still don’t know what s is! Need to relate it to slant range (straight distance a radar beam would traverse over a given time), or relate h to slant range without requiring knowledge of s… • Use trigonometry (and a chalk board)! • Use similar argument to find s as a function of slant range (homework problem)…

24. Refraction Doviak and Zrnic (1993), p. 23

25. Refraction • Sadly (actually, happily), the atmosphere is typically not “standard”. Generally get four regimes of refraction: • (1) Normal refraction:Here, the standard “lapse rate” applies for the index of refraction (use R’=4R/3)

26. Refraction • Sadly (actually, happily), the atmosphere is typically not “standard”. Generally get four regimes of refraction: • (2) Subrefraction:Not very common – occurs sometimes in the desert with superadiabatic lapse rates near the surface, and in forest fires • Results in underestimation of echo tops • The beam is not bent as far toward the earth’s surface as it moves away from the radar

27. Refraction • Sadly (actually, happily), the atmosphere is typically not “standard”. Generally get four regimes of refraction: • (3) Superrefraction:Very common during temperature inversions, or sharp vertical decreases in moisture (nocturnal and BL inversions, WAA, fronts, outflow boundaries, etc.) • Results in overestimation of echo tops, increased ground clutter • The beam is bent farther toward the earth’s surface as it moves away from the radar

28. Refraction • Sadly (actually, happily), the atmosphere is typically not “standard”. Generally get four regimes of refraction: • (4) Ducting:Common during very strong temperature inversions, or vertical decreases in moisture (nocturnal and BL inversions, WAA, fronts, outflow boundaries, etc.) • Results in enhanced signature close to the radar; produces a “radar hole”  a region between the lowest and highest elevation angle that is not observed by the radar • The beam at sufficiently low elevation angles is bent toward the earth’s surface and intersects it (i.e., never rises above a particular height above the ground)

29. Refraction • Sadly (actually, happily), the atmosphere is typically not “standard”. Generally get four regimes of refraction: • Subrefraction, superrefraction, and ducting are collectively referred to as anomalous propagation. http://www.srh.noaa.gov/jetstream/doppler/ap_max.htm http://www.letxa.com/anomalyap.php

30. Refraction • An example of ducting, from Doviak and Zrnic (1993) Radar hole may be found here In cases of ducting, the range of the near for low elevation angles may be very large. Useful for tracking surface objects (military advantages).