1. Atmospheric InstrumentationM. D. Eastin Radar Equation and Reflectivity Φ rΦrΦ c τ /2

2. Atmospheric InstrumentationM. D. Eastin Outline Radar Equation and Reflectivity Basic Approach to Radar Equation Development Solitary Target Power incident on target Power scattered back toward the radar Power received by the antenna Distributed (Multiple) Targets Distributed (Multiple) Weather Targets Equivalent Radar Reflectivity Factor

3. Atmospheric InstrumentationM. D. Eastin Radar Equation Development Basic Approach: Conceptual Development Radar Observation:The “return echo power” scattered back from “targets” can provide useful information about the target’s characteristics ( size / number of aircraft → enemy bomber raid? ) ( size / number of raindrops → reflectivity → storm structure ) Radar Equation:Provides a relationship between (1) the return echo power, (2) the target’s characteristics, and (3) unique antenna/radar characteristics Basic development is common to all radars! Solitary Target: Develop radar equation for a single target (i.e., one raindrop) 1. Determine the transmitted power flux density incident on the target 2. Determine the power flux density scattered back toward the radar 3. Determine the amount of back-scattered power collected by the radar antenna Distributed Targets:Expand to allow for multiple targets with the volume

4. Atmospheric InstrumentationM. D. Eastin Radar Equation Development Basic Approach: Notation → Decibel Differences between transmitted pulse power (P T = 1 MW = 10 6 W) and the full range of return echo power (P R = 10 -15 W up to 10 -6 W) are several orders of magnitude A convenient method to express such large ranges is using decibel (dB) notation: where: P OBS =observed power (W) P REF = constant reference power (W) In practice, two radar parameters are expressed in decibels → each discussed later in detail Return Echo Power Radar Reflectivity P REF =1 ×10 -3 WZ REF =1 mm 6 m -3 =1 mW=Z for one 1-mm drop

5. Atmospheric InstrumentationM. D. Eastin Radar Equation – Solitary Target Transmitted Power: Isotropic Radar:Transmits equal power in all spherical directions with a power flux density (S ISO ) at a given range: (1) where:S=power flux density (W m -2 ) P T =total transmitted power (W m -2 ) r=range from antenna (m) Directional Radar:Most radars attempt to focus all transmitted power into a narrow beam This is NOT an easy task, but most radars come close Gain Function:Gain is the ratio of the power flux density at range r / azimuth θ / elevation φ for a directional antenna, to the power flux density for an isotropic antenna transmitting the same total power: (2) where:G=gain (no units)

6. Transmitted Power: Gain Function:If we combine (1) and (2), we can solve for the beam’s power flux density: (3) No radar emits a perfect conical beam due to manufacturing imperfections in the waveguide and antenna → gain functions are unique to each radar Main Lobe:Maximum gain on the order of 40-50 dB ** (see two slides prior) Side Lobes:Maximum gain of 4 dB (most are less than 0 dB) Back Lobe:Maximum gain less than 0 dB Effective beam width (Θ): Defined at the location equivalent to 3 dB less than the peak gain of the main lobe → narrower beam widths are desired Atmospheric InstrumentationM. D. Eastin Radar Equation – Solitary Target Main Lobe Side Lobes Back Lobe 3 dB Θ

7. Transmitted Power: Antenna Size: Since effective beam width and main lobe power are linked to antenna size, so is the gain: (4) where: A E =effective antenna area (m 2 ) λ = transmitting wavelength (m) Large Antenna→Large gain →Large wavelengths →Small beam widths →Desired →Budget? Small Antenna→Small gain →Small wavelengths →Large beam widths →Less desired →Budget? Atmospheric InstrumentationM. D. Eastin Radar Equation – Solitary Target 10 cm 3 cm

8. Atmospheric InstrumentationM. D. Eastin Radar Equation – Solitary Target Problems Associated with Side Lobes: Any return echo from a side lobe is interpreted as a weak return from the main lobe which effectively produces a three-dimensional expansion of the storm size Horizontal Spreading of Storm Vertical Spreading of Storm Top

9. Atmospheric InstrumentationM. D. Eastin Radar Equation – Solitary Target Primary Method to Minimize Side Lobes: Use a parabolic antenna Parabolic antennas allow for “tapered illumination” which minimize the transmitted power flux density along the edges Effects of Tapered Illumination: 1. Reduction of side lobe returns 2. Reduction of maximum gain 3. Increased beam width The last two are undesirable, but in practice, parabolic antennas reduce side lobes by ~80%, reduce gain by less than 5%, and increase beam width by less than 25% → acceptable compromise Beam Geometry Outgoing Power Flux Density

10. Atmospheric InstrumentationM. D. Eastin Radar Equation – Solitary Target Radar Cross Section (σ): Defined as the ratio of the power flux density scattered by the target in the direction of the antenna to the power flux density incident on the target (both measured at the target radius) (5) NOTE:S BACK is not measured at a radius S BACK is measured at the radar For practical purposes, the radar cross section is redefined as the power flux density received at a point on a spherical surface (6) Radar Transmitted Power Flux Density Target Backscatter Power Flux Density

11. Atmospheric InstrumentationM. D. Eastin Radar Equation – Solitary Target Radar Cross Section (σ): The observed radar cross section of a “target” depends on: 1. Target shape 2. Target size relative to the radar’s wavelength (more on this later…) 3. Complex dielectric constant of the target (more on this later…) 4. Viewing aspect from the radar Fighter AircraftCommercial Aircraft

12. Atmospheric InstrumentationM. D. Eastin Radar Equation – Solitary Target Received Power at the Antenna: Recall from before: (3)(4) (6) Substituting (3) into (6): (7) We can now define the received power (P R ) at the antenna as: (8) Power flux density incident on target Gain – Antenna size relationship Radar cross section Power flux density of a target’s backscatter received at the antenna Power flux density incident on a target

13. Atmospheric InstrumentationM. D. Eastin Radar Equation – Solitary Target Received Power at the Antenna: Substituting (4) and (7) into (8) yields: (9) After re-arranging: Written in terms of antenna area: Radar equation for a single isolated target (e.g. an airplane, bird, or one raindrop) Constant Radar Characteristics Target Characteristics Constant Radar Characteristics Target Characteristics What do these equations tell us about radar returns from a single target?

14. Atmospheric InstrumentationM. D. Eastin Radar Equation – Distributed Targets Distributed Target: A target consisting of multiple scattering objects All raindrops intercepted by a single radar pulse Contributing Region: Total volume containing all objects from which back-scattered return power arrives back at the radar simultaneously Single Pulse Volume: First, assuming the main lobe pulse is cylindrical at large distances (>10 km) from the radar Volume of contributing region for a single pulse: (10) Cross-sectional area of a pulse at radius ( r ) and angular beam width ( Φ ) Φ rΦrΦ c τ /2 Contributing length of a pulse transmitted for duration ( τ )

15. Atmospheric InstrumentationM. D. Eastin Radar Equation – Distributed Targets Single Pulse Volume – An Example: WSR-88D Radar Let’s assume a radar observes a light rain shower ~100 km away: Pulse duration / width ( τ ) =1.57 μs Angular beam width (1º)=0.0162 radians Range from radar (r) =100 km If the raindrop concentration in this volume was one drop per cubic meter (1 drop m -3 ), then a single pulse volume contains: 520 million raindrops!!

16. Atmospheric InstrumentationM. D. Eastin Radar Equation – Distributed Targets Accounting for Gain Shape: The gain function’s main lobe is not truly cylindrical (but it’s a good first approximation) Rather, the main lobe exhibits a Gaussian shape whereby it maximizes along the beam axis and decreases with angular distance from the axis Accounting for such shape can be accomplished by incorporating a simple geometric correction→ (11) factor [ 1 / 2ln(2) ] into the pulse volume definition Mean Radar Cross Section (σ AVG ): Each radar pulse volume can contain an array of various sized /shaped objects Their collective cross section is a mean of all individual cross sections in the pulse volume For simplicity, we will use the standard algebraic mean (12) Main Lobe Θ

17. Atmospheric InstrumentationM. D. Eastin Radar Equation – Distributed Targets Received Power at the Antenna: We can now multiple the radar equation for a single target (9) by the pulse volume (11) and then substitute the mean radar cross section (12) to arrive at a radar equation describing the received power at the antenna for a distribution of targets within any pulse volume… (9)(11)(12) …and after re-arranging: (13) Constant Radar Characteristics Target Characteristics

18. Atmospheric InstrumentationM. D. Eastin Radar Equation – Distributed Weather Targets Modifying our Radar Equation for Weather Targets: Meteorologists are interested in weather targets, so we can develop a special form of the distributed radar equation for typical collections of precipitation particles Airports are interesting in tracking local aircraft, so special forms of the radar equation can be developed to better detect commercial and recreational aircraft Militaries are interested in tracking aircraft and ships (friendly and enemy), so special forms of the radar equation can be developed to better detect (stealth) aircraft and military ships. Four tasks must be completed: 1. Understand the impact of a radar pulse’s electric field on a water particle 2. Find the radar cross section for a single precipitation particle 3. Find the total radar cross section for the entire contributing region 4. Obtain the average radar reflectivity from all particles in that region

19. Atmospheric InstrumentationM. D. Eastin Radar Equation – Distributed Weather Targets First Assumption: All targets are spheres! Small Raindrops=Spheres Ice Crystals=Variety of shapes Large raindrops=Ellipsoids Graupel / Hail=Variety of shapes

20. Atmospheric InstrumentationM. D. Eastin Radar Equation – Distributed Weather Targets Second Assumption: All targets are small! If targets are sufficiently small compared to the wavelength of the transmitted radar pulse, then the backscatter can be described by Rayleigh Scattering Theory Types of scattering: Rayleigh Mie Optical How small? Why Raleigh scattering? Target radius less than λ /20 Since the target particles are much smaller than the variability associated with the radar pulse’s electric-field (a sine wave), then we can assume the electric-field across the particle will be uniform

21. Atmospheric InstrumentationM. D. Eastin Radar Equation – Distributed Weather Targets Impact of a Radar Pulse’s Electric Field on a Water Particle: The radar pulse (and its associated electric field) will induce an electric dipole within any homogeneous dielectric sphere (i.e., a water drop or ice sphere) Induced electric dipole vector ( E DP )→ Direction is the same as the pulse’s electric field → Magnitude is the product of the incident electric field and the polarization of the sphere where: ε 0 =permittivity in a vacuum K = dielectric constant for water / ice D = diameter of a spherical particle (m) E BEAM = amplitude of incident electric field (W) The sphere then scatters that portion of the electric field equivalent to the dipole magnitude such that the electric backscatter received at the radar is defined by: OR(14)

22. Atmospheric InstrumentationM. D. Eastin Radar Equation – Distributed Weather Targets Radar Cross Section of a Single Small Dielectric Sphere: The relationship between the power flux density (S) and the electric field (E) for both the transmitted pulse and that received back at the radar are related via: (15) Combining (14) and (15) with (6) we get the radar cross section for a single sphere:... (6)(14) …and after re-arranging: (16) What does this equation tell us about radar returns from a single weather target?

23. Atmospheric InstrumentationM. D. Eastin Radar Equation – Distributed Weather Targets Radar Cross Section for Multiple Small Dielectric Spheres: Following the same methods as before, we can combine (12) and (16) such that… (12)(16) …the mean radar cross section for an array of spherical water particles: (17) We can now define a mean radar reflectivity factor for all spherical water particles in the radar pulse volume: (18)OR Equation 3.1 in the Fabry text

24. Atmospheric InstrumentationM. D. Eastin Radar Equation – Distributed Weather Targets Received Power at the Antenna: Using (18), we can now substitute the mean radar cross section for an array of spherical water particles (17) into the generic radar equation for distributed targets (13) to arrive at a radar equation describingthe received power at the antenna for a distribution of weather targets within any pulse volume… (13) (17) (18) …and after re-arranging: (19) Constant Radar Characteristics Target Characteristics

25. Atmospheric InstrumentationM. D. Eastin What is the Dielectric Constant (K)? A measure of the scattering and absorption properties of a medium (water or ice) when interacting with an electromagnetic field (a radar beam pulse) where: Permittivity in a medium Permittivity in a vacuum Dielectric Constant Values – Water Targets LIQUID: K 2 =0.930 (spheres) ICE: K 2 =0.176 (solid spheres) K 2 =0.202 (snow flakes) Since K 2 varies as a function of particle shape and water phase, we would need to know the shape and phase of every water particle present in the pulse volume! No way to obtain this information! We must make an assumption about the K 2 value! Equivalent Radar Reflectivity Factor

26. Atmospheric InstrumentationM. D. Eastin Equivalent Radar Reflectivity Factor A Practical Parameter – Equivalent Radar Reflectivity (Z E ) While the radar equation was developed under assumptions that any return echo was directly related to the number, size, shape, and phase of small water particles contained within the radar pulse volume, in practice we cannot be certain about the following: 1. Target type (water particles, insects, birds, airplanes?) 2. Target size (Rayleigh scattering?) 3. Origin of total return echo power (main lobe, side lobe, other sources?) 4. Nature of the desired water targets (particle shape and/or phase?) The radar equation used in practice assumes constant K 2 = 0.93 (spherical liquid drops) (20) Constant Radar Characteristics Target Characteristics

27. Atmospheric InstrumentationM. D. Eastin Equivalent Radar Reflectivity Factor A Practical Parameter – Equivalent Radar Reflectivity (Z E ) Finally, we can solve the practical radar equation for distributed weather targets (20) for the equivalent radar reflectivity (ZE (21) The equivalent reflectivity factor (Z E ) is computed by the radar processing software based on (1) measured return echo power (P R ), (2) range (r) – determined from the time between pulse transmission and echo return, and (3) known constants and radar characteristics The computed Z E are then transformed into decibel notation for display: (22) Constant Radar Characteristics Target Characteristics

28. Atmospheric InstrumentationM. D. Eastin Equivalent Radar Reflectivity Factor IMPORTANT – Decibel Notation (dB) and Equivalent Radar Reflectivity (Z E ) Equation (21) provides Z E in units of m 6 / m 3 Equation (22) provides dBZ using Z E in units of mm 6 / m 3 (so the fraction is unit-less) The computed Z E from (21) must be multiplied by 10 18 to obtain a correct dBZ! (21) (22)

29. Atmospheric InstrumentationM. D. Eastin Radar Equation Development IMPORTANT – Review of Assumptions: 1. All precipitation particles are homogeneous dielectric spheres with diameters small compared to the radar wavelength (the Rayleigh approximation) 2. All particles are evenly spread through the contributing region 3. The equivalent reflectivity factor (Z E ) is uniform throughout the contributing region and constant during the time period required to obtain the mean value of the received power 4. All particles have the same dielectric constant – assumed to be liquid water spheres 5. The main lobe of the radar pulse is adequately described by a Gaussian function 6. Microwave attenuation between the radar and the target is negligible 7. Multiple scattering is negligible 8. The incident and backscattered pulses are linearly polarized.

30. Atmospheric InstrumentationM. D. Eastin Radar Equation Development IMPORTANT – Validity of the Rayleigh Approximation? Valid: Invalid: λ = 10 cm Raindrops: 0.01 – 0.5 cm (all rain) Ice crystals: 0.01– 3 cm (all snow) Ice stones: 0.5 – 2.0 cm (small to moderate hail) λ = 3 cm Raindrops: 0.01 – 0.5 cm (all rain) Ice crystals: 0.01– 0.5 cm (single crystals) Ice stones: 0.1 - 0.5 cm (graupel) λ = 10 cm Ice stones: > 2 cm (large hail) λ = 3 cm Ice crystals: > 0.5 cm (snowflakes) Ice stones: > 0.5 cm (hail and large graupel)

31. Atmospheric InstrumentationM. D. Eastin Summary Radar Equation and Reflectivity Basic Approach to Radar Equation Development Solitary Target Power incident on target Power scattered back toward the radar Power received by the antenna Distributed (Multiple) Targets Distributed (Multiple) Weather Targets Equivalent Radar Reflectivity Factor

32. Atmospheric InstrumentationM. D. Eastin References Atlas, D., 1990: Radar in Meteorology, American Meteorological Society, 806 pp. Crum, T. D., R. L. Alberty, and D. W. Burgess, 1993: Recording, archiving, and using WSR-88D data. Bulletin of the American Meteorological Society, 74, 645-653. Doviak, R. J., and D. S. Zrnic, 1993: Doppler Radar and Weather Observations, Academic Press, 320 pp. Fabry, F., 2015: Radar Meteorology Principles and Practice, Cambridge University Press, 256 pp. Reinhart, R. E., 2004: Radar for Meteorologists, Wiley- Blackwell Publishing, 250 pp.