Electromagnetic Waves and Their Propagation Through the Atmosphere

ELECTRIC FIELD An Electric field exists in the presence of a charged body ELECTRIC FIELD INTENSITY (E) A vector quantity: magnitude and direction (Volts/meter) MAGNITUDE OF E: Proportional to the force acting on a unit positive charge at a point in the field DIRECTION OF E: The direction that the force acts

The Electric Field (E) is represented by drawing the Electric Displacement Vector (D), which takes into account the characteristics of the medium within which the Electric Field exists. e, the Electric Conductive Capacity or Permittivity, is related to the ability of a medium, such as air to store electrical potential energy. Vacuum: Air: Ratio:

The Electric Displacement Vector, D, is used to draw lines of force. Units of D:

MAGNETIC FIELD A Magnetic field exists in the presence of a current MAGNETIC FIELD INTENSITY (H) A vector quantity: magnitude and direction (amps/meter) MAGNITUDE OF H: Proportional to the current DIRECTION OF H: The direction that a compass needle points in a magnetic field

The Magnetic Field (H) is represented by drawing the Magnetic Induction Vector (B), which takes into account the characteristics of the medium within which the current flows. m, the Magnetic Inductive Capacity, or Permeability, is related to the ability of a medium, such as air, to store magnetic potential energy. Vacuum: Air: Ratio:

Magnetic Fields: Magnetic fields associated with moving charges (electric currents) I: Current B: Magnetic Induction Magnetic Field Lines are closed loops surrounding the currents that produce them

Maxwell’s Equations for time varying electric and magnetic fields in free space Simple interpretation Divergence of electric field is a function of charge density A closed loop of E field lines will exist when the magnetic field varies with time Divergence of magnetic field =0 (closed loops) A closed loop of B field lines will exist in The presence of a current and/or time varying electric field (where ρ is the charge density)

Electromagnetic Waves: A solution to Maxwell’s Equations Electric and Magnetic Force Fields Propagate through a vacuum at the speed of light: Electric and Magnetic Fields propagate as waves: or: where: ρ, θ, φ are coordinates, A is an amplitude factor, ν is the frequency and Ψ is an arbitrary phase

Electromagnetic waves: Interact with matter in four ways: Reflection: Refraction:

Scattering: Diffraction:

Electromagnetic Waves are characterized by: Wavelength, λ [m, cm, mm, mm etc] Frequency, ν [s-1, hertz (hz), megahertz (Mhz), gigahertz (Ghz) where: c = λν

Time variations in charge, voltage and current in a simple Dipole Antenna Pt. A Pt. B wavelength All energy stored in electric field All energy stored in magnetic field Energy is 1) stored in E, B fields, 2) radiated as EM waves, 3) Dissipated as heat in antenna Near antenna: Energy stored in induction fields (E, B fields) >> energy radiated (near field) More than a few λ from antenna: Energy radiated >> energy stored in induction fields (far field)

Polarization of electromagnetic waves The polarization is specified by the orientation of the electromagnetic field. The plane containing the electric field is called the plane of polarization.

For a monochromatic wave: Electric field will oscillate in the x,y plane with z as the propagation direction For a monochromatic wave: where f is the frequency and d is the phase difference between Exm and Eym and the coordinate x is parallel to the horizon, y normal to x, and z in the direction of propagation. If Eym = 0, Electric field oscillates in the x direction and wave is said to be “horizontally polarized” If Exm = 0, Electric field oscillates in the y direction and wave is said to be “vertically polarized” If Exm = Eym, and d = π/2 or - π/2, electric field vector rotates in a circle and wave is circularly polarized All other situations: E field rotates as an ellipse

Propagation of electromagnetic waves in the atmosphere Speed of light in a vacuum: Speed of light in air: Refractive index: At sea level: n = 1.0003 In space: n = 1.0000 Radio refractivity: At sea level: N = 300 In space: N = 0

The Refractive Index is related to: Density of air (a function of dry air pressure (Pd), temperature (T), vapor pressure (e) 2. The polarization of molecules in the air (molecules that produce their own electric field in the absence of external forces) The water molecule consists of three atoms, one O and two H. Each H donates an electron to the O so that each H carries one positive charge and the O carries two negative charges, creating a polar molecule – one side of the molecule is negative and the other positive.

In the atmosphere, n normally decreases continuously with height… Snell’s law: n - Dn n i r Vi Vr Where: i is the angle of incidence r is the angle of refraction Vi is the velocity of light in medium n Vr is the velocity of light in medium n - Dn In the atmosphere, n normally decreases continuously with height… Therefore: due to refraction, electromagnetic rays propagating upward away from a radar will bend toward the earth’s surface

Earth curvature Electromagnetic ray propagating away from the radar will rise above the earth’s surface due to the earth’s curvature.

Ray Path Geometry Consider the geometry for a ray path in the Earth’s atmosphere. Here R is the radius of the Earth, h0 is the height of the transmitter above the surface, φ0 is the initial launch angle of the beam, φh is the angle relative to the local tangent at some point along the beam (at height h above the surface at great circle distance s from the transmitter).

Equation governing the path of a ray in the earth’s atmosphere: (1) where R is the radius of the earth, h is the height of the beam above the earth’s surface, and s is distance along the earth’s surface. To simplify this equation we will make three approximations 1. Large earth approximation 2. Small angle approximation 3. Refractive index ~ 1 in term:

1 1/R 1 X X X Approximate equation for the path of a ray at small angles relative to the earth’s surface: (2) Or, in terms of the elevation angle of the beam

Spherically Stratified Atmosphere; Ray Path Equation Integrating (2) yields, (dh/ds)2 = 2∫ (1/R + dn/dh) dh + constant (3) Since dh/ds ≈ φ for small φ, (3) can be written as, 1/2(φh2 - φ02) = (h - h0)/R + n - n0 = (h/R + n) - (h0/R + n0) Letting M = [h/R + (n-1)] x 106, we have = (M - M0)10-6 M is the so-called modified index of refraction. M has a value of approximately 300 at sea level.

Curvature of Ray Paths Relative to the Earth If the vertical profile of M is known (say through a sounding yielding p, T and q), φh can be calculated at any altitude h, that is, the angle relative to the local tangent. Lets now consider the ray paths relative to the Earth. For the case of no atmosphere, or if N is constant with height (dN/dh = 0), the ray paths would be straight lines relative to the curved Earth. dφ/ds = 1/R + dn/dh 1/R for n constant with height (No atmosphere case?) (“Flat earth” case?) For n varying with height, dφ/ds = 1/R + dn/dh < 1/R since dn/dh < 0 For the special case where dn/dh = -1/R, dφ/ds = 0. Hence the ray travels around the Earth concentric with it, at fixed radius, R + h. This is the case of a trapped wave. “DUCTING”

Curvature of Ray Paths Relative to the Earth For convenience, it is is easier to introduce a fictitious Earth radius, 1/R’ = 1/R + dn/dh For typical conditions, dn/dh = -1/4 R m-1 Hence R’ = R/(1 - 1/4) = 4/3 R This is the effective Earth radius model, to allow paths to be treated as straight lines. Doviak and Zrnic (1993) provide a complete expression for h vs. r, where r is the slant range (distance along the ray). h = {r2 + (keR)2 + 2rkeRsinθ}1/2 - keR where h is beam height as slant range r, θ is the elevation angle of the antenna, and ke is 4/3 (R is the actual Earth radius).

Curvature of Ray Paths Relative to the Earth An additional equation of interest is the equation that provides the great circle distance s, from the radar, for the r, h pair (slant range, beam height), which is s = keR sin-1[rcosθ/(keR + h)] We can get even simpler and consider a the height of the beam at slant range R and elevation angle θ, h (km) = R2/17000 + R sin θ R h θ

So, let ae= 4/3 a; then for convenience of computation: Use standard atmosphere, solve Diff. Eq. describing ray path for height of beam above surface of earth (assumes dn/dh is small): d2h/ds2 – (2/R + 1/n * dn/dh)(dh/ds)2 – (R/a)2(1/R + 1/n * dn/dh) = 0 Where: a= earth radius; s= arc distance; h= height above earth surface n= refractive index; R= h + a; r= slant range along beam Physically: Via equation for refractivity, we expect the beam to bend toward the surface since dP,e/dz < 0 and < dT/dz. However, h increases with s due to 1/R (curvature of earth’s surface, which diverges from beam position). DEQ above expresses this relationship as it relates earth’s geometry and the assumed refraction of the standard atmosphere to beam height and arc distance. Doviak and Zrnic (1993) Sec. 2.2 show how this can be reduced to two equations for h and s using the 4/3 Earth radius model (4/3 Earth radius - dn/dh assumed to be constant - of order 0.25/a) So, let ae= 4/3 a; then for convenience of computation: h=[r2 + (ae)2 + 2raesinΦ0]1/2 – ae s=aesin-1(rcosΦ0/[ae+h]) STANDARD REFRACTION: What we expect the beam to do over the curved surface of the earth Φ0 h s r

4/3 Earth Radius Model for Beam Propagation (Standard Refraction/Reference Atmosphere Assumed) h=[r2 + (ae)2 + 2raesinθe]1/2 – ae S=aesin-1(rcosθe/[ae+h]) Θe = elevation angle To get h as a f(slant range:R), which is measured by the radar, use this simple formula: h (km)= R2/17000 + R sinθe (with R in km) Doviak and Zrnic (1993)

Non-Standard Refraction Non-standard refraction typically occurs with the temperature distribution does not follow the standard lapse rate (dn/dh ≠ -1/4 (R)). As a result, radar waves may deviate from their standard ray paths predicted by the previous model. This situation is known as abnormal or anomalous propagation (AP). Abnormal downward bending ------- super-refraction (most common type of AP) Abnormal upward bending ----------- sub-refraction Super-refraction is associated most often with cold air at the surface, giving rise to a near surface elevated temperature inversion in which the T increases with height. Most commonly caused by radiational cooling at night, or a cold thunderstorm outflow. Since T increases with height, n decreases (rapidly) with height (dn/dh is strongly negative). Since n = c/v, v must increase with height, causing downward bending of the ray path.

Recall Snell’s Law: n1sinθ1 = n2sin θ2 n1 n2 θ1 θ2 > v2/v1 = sinθ2/sinθ1 v2 > v1 Wave (beam) is bent downward (refracted) in the atmosphere So relative to the refractivity, what’s important here? dN/dZ – change in refraction with height- this causes velocity differences across the beam. 4 cases of refraction (dN/dZ): Standard: dN/dZ ~ 0 and -40 km-1 Super: dN/dZ < -79 km-1 and > -158 km-1 Sub: dN/dZ > 0 Ducting: dN/dZ < -158 km-1 (dn/dh = -1/R) Non-Standard

dN/dZ < -79 km-1 and > -158 km-1 Non-Standard Refraction Super-Refraction (most common) dN/dZ < -79 km-1 and > -158 km-1 h h’ Φ0 Beam is bent downward more than standard Situations: Temperature inversions (warm over cold air; stable layers) Sharp decrease in moisture with height And (2) can occur in nocturnal and trade inversions, warm air advection (dry), thunderstorm outflows, fronts etc. Result: Some increased clutter ranges (side lobes) Overestimate of echo top heights (antenna has to be tilted higher to achieve same height as standard refracted beam)- see figure above Most susceptible at low elevation angles (e.g., typically less than 1o)

Sub-Refraction (not as common) dN/dZ > 0 km-1 DP T Inverted-V sounding h h’ Φ0 Beam is bent upward more than standard Situations: Inverted-V sounding (typical of desert/intermountain west and lee-side of mountain ranges; microburst sounding; late afternoon and early evening; see figure) Result: Underestimate of echo top heights (beam intersects top at elevation angles lower than in standard refraction case)- see figure above Most susceptible at low elevation angles (e.g., typically less than 1o)

Ducting or Trapping (common) dN/dZ < -158 km-1 Beam is severely bent downward and may intersect the surface (especially at elevation angles less than 0.5o) or propagate long distances at relatively fixed heights in an elevated “duct”. Situations: Strong temperature inversions (surface or aloft) Strong decreases in moisture with height Result: Markedly Increased clutter ranges at low elevation angles Range increases to as much as 500% in rare instances (useful for tracking surface targets) Most susceptible at low elevation angles (e.g., typically less than 1o) Elevated ducts can be used as a strategic asset for military airborne surveillance and weapons control radars. E.g., if a hostile aircraft is flying in a ducting layer … it could be detected a long way away, while its radar cannot detect above or below the ducting layer. Conversely, friendly aircraft may not want to be located in the duct.

Example of ray paths in surface ducting Modeled with 100 m deep surface inversion with dN/dz=300 km-1 and standard thereafter. Example of ray paths in surface ducting Doviak and Zrnic (1993) One moral of the whole refraction story……..knowing the exact location of the beam can be problematic. Remember this when you have the opportunity to compare the measurements of two radars supposedly looking at the same storm volume!

Inversions with dN/dZ < -158 km-1 lead to anomalous propagation Most common on clear nights during the early morning hours. Largely dissipates by midday. Common over water, especially in the cold season. Radar beam is bent into the ground and returns a strong signal to radar. Radar echoes are NOT real, there was no precipitation occurring in the image at right. Source: Meteorological Service of Canada

Scanning strategies for scanning radars must take into account the propagation path of the beam if certain operational or scientific objectives are to be addressed. Here, 3 common NWS NEXRAD Volume Coverage Patters (VCPs) are illustrated. NEXRADs have a 5-6 minute scan update requirement for severe weather detection, so they vary their VCPs and scan rates depending on the weather situation. VCP 31 “clear air mode” 6 min update, slow scan rate VCP 11 “severe weather mode” 5 min update, fast scan rate VCP 21 Wide-spread precip 6 min update, slow scan rate

Big implication of radar beam height increasing with range (under normal propagation conditions) combined with broadening of the radar beam: The radar cannot “see” the low level structures of storms, nor resolve their spatial structure as well as at close ranges. Thus, for purposes of radar applications such as rainfall estimation, the uncertainty of the measurements increases markedly with range. Storm 1 Storm 2

Beam Blockage in Complex Terrain Beam propagation is a function of the vertical refractivity gradient (dN/dz) N = 77.6(p/T) - 5.6(e/T) + 3.75x105(e/T2) dN/dz is sensitive to p, T, e Thus, changes in the vertical profiles of these quantities can change the height of the ray path as it propagates away from the radar This is especially important in complex terrain, because the amount of beam blockage will change depending on the vertical refractivity gradient dN/dZ = -40/km dN/dZ = -80/km ) 42

Radar Rainfall Climatology - KPBZ Warm Season Cold Season 43

Mid-Atlantic River Forecast Center (MARFC) Height of Lowest Unobstructed Sampling Volume Radar Coverage Map 44

West Gulf River Forecast Center (WGRFC) Height of Lowest Unobstructed Sampling Volume Radar Coverage Map 45

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PRECIPITATION MOSAIC RADAR COVERAGE MAP 47

Northwest River Forecast Center (NWRFC) Warm Season Cold Season 48

Say you’d like to site a radar for a research experiment. In a perfect world…you’d like to be able to take a swim after work, but AP and beam blockage may be a problem. Sidelobes may intersect the highly reflective ocean – creating “sea clutter” 49

Mountains can be a problem… 0.5° 1.5° 50

Local effects can be a problem too – topographic maps and DEMs can help, but still need to conduct a site survey to see trees, antennas, buildings, and overpasses. 51

Often times you end up in places like this… 52

Height of a ray due to earth’s curvature and standard atmospheric refraction

Anomalous Propagation The propagation of a radar ray along a path other than that associated with standard atmospheric refraction

Anomalous propagation occurs when the index of refraction decreases rapidly with height in the lowest layers of the atmosphere Recall that the Refractive Index is related to: n decreases rapidly when T increases with height and/or e decreases with with height in the lowest layer

Effects of anomalous propagation: Note cell towers along Interstates! Note buildings In Champaign, IL

Propagation of Electromagnetic Waves In this section we will discuss the propagation of EM waves including further discussions on the index of refraction, Snell’s Law, and derivation of equations for the ray path of a radar wave traveling under various atmospheric conditions. Since the atmosphere is a non-vacuum, we deal with wave speeds that are different from the speed of light, c = 2.998 x 108 m/s. As discussed in the previous section, the wave speed for a non-vacuum defines the index of refraction, n = c/v where v is the wave speed in the particular medium. Since c = √ε0μ0 and v = √ε1μ1, we have n2 = εμ where ε = ε 1/ε0 and μ = μ1 /μ0 Since μ is approximately 1 for most media considered, n2 = ε. With ε>1, n>1 and hence v<c (by a small amount). The general form of the index of refraction is of the form m = n - ik where k is the absorptivity of the medium.

Propagation of Electromagnetic Waves Index of refraction for the atmosphere governs the path of radar waves The atmosphere is an inhomogeneous medium, with variations in temperature, pressure and water vapor, all of which contribute to changes in the index of refraction. Index of refraction for dry air, or N, the refractivity For dry air, N = (n-1)106 = K1p/T where P is in mb, T in °K, K1 = 77.6 (°K/mb) Substituting from the Ideal Gas Law, (n-1) 106 = K1Rρ = constant x ρ Therefore, dn/dz ≈ dρ/dz

Propagation of Electromagnetic Waves Water vapor contribution to the index of refraction, n Since air molecules essentially have no permanent dipole moment, N (dry air) does not vary with frequency. However, this is not the case for the water vapor molecule, which has a permanent dipole moment. The degree of alignment of this dipole moment with the incident E field vector is frequency dependent. For microwave frequencies, N = (n-1)106 = K3e/T2 - K2e/T where e is the vapor pressure in mb; K2 = 5.6 °K/mb; K3 = 3.75x105 (°K)2/mb Index of refraction may be found by adding components for both dry air and water vapor, N = K1p/T + K3e/T2 - K2e/T Key question: How does N vary with height and with varying atmospheric conditions?

Snell’s Law Curvature of ray path due to “n” changes, relative First examine simple refraction in terms of Snell’s Law Since p and e decrease exponentially with height, n decreases with altitude (these affects offset the linear decrease in height for T, for most situations). Since n = c/v, v increases with height and hence the wave is bent downward. Snell’s Law is n-Δn r sin i/sin r = vi/vr vr vi i n since vr> vi it follows that sin r > sin i and hence r > i This is the typical situation for a ray path in the atmosphere under conditions where the temperature decreases with height. Curvature of ray path due to “n” changes, relative to curvature of Earth is key issue!

Spherically Stratified Atmosphere; Ray Path Equation For dn/dh small, Hartee, Michel and Nicolson (1946) derived an exact differential equation for a radar ray path in a spherically-stratified atmosphere. d2h/ds2 - (2/(R+h) + 1/n(dn/dh))(dh/ds)2 -( (R+h)/R)2 (1/(R+h) + 1/n(dn/dh)) = 0 (1) where d2h/ds2 is the curvature of the ray path. Under most conditions, the following assumptions can be made: (dh/ds)2 << 1 n ≈ 1 h << R With these assumptions, (1) reduces to d2h/ds2 = 1/R + dn/dh (2)