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Radar Equations Radar Meteorology M. D. Eastin Radar Equations Outline • Basic Approach to Radar Equation Development • Solitary Target • Power incident on target • Power scattered back toward radar • Amount of power collected by the antenna • Distributed (Multiple) Targets • Distributed (Multiple) Weather Targets Radar Meteorology M. D. Eastin Radar Equation Development Basic Approach: Radar Observation: Measurement of echo power received from a target provides useful information about the target (# raindrops ~ reflectivity) Radar Equation: Provides a relationship between the received power, the characteristics of the target, and the unique characteristics of the antenna/radar design 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 per unit area (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 actually collected by the radar Distributed Targets: Radar Meteorology Expand to allow for multiple targets with the volume M. D. Eastin Transmitted Power Isotropic Antenna: • An isotropic radar transmits power equally in all directions • Power flux density at a given range from an isotropic antenna is: Siso Pt 4r 2 (1) where S = power flux density (W/m2) Pt = transmitted power (W/m2) r = range from antenna Radar Meteorology M. D. Eastin Transmitted Power Directional Antenna: • Most radars attempt to focus all of the transmitted power into a narrow window (a beam) • This is NOT an easy task, but most radars come close Gain Function: • Gain is the ratio of the power flux density at radius r, azimuth θ, and elevation φ for a directional antenna to the power flux density for an isotropic antenna transmitting the same total power: G( , ) Sinc ( , ) Siso (2) • Combining (2) with (1): S inc GPt 4r 2 (3) • In practice, it also incorporates absorptive losses by the antenna and waveguide • The gain function is unique for each radar! Radar Meteorology M. D. Eastin Transmitted Power What does the Gain Function look like? Main Lobe 3D Depiction of Gain (dB) Side Lobes Radar Meteorology M. D. Eastin Transmitted Power What does the Gain Function look like? 2D Depiction of Gain (dB) • Main lobe (i.e. the beam) has a maximum gain of 30 dB • Strongest side lobes are ~4 dB with the majority less than 0 dB • All back lobes are less than 0 dB Effective beam width (Θ) defined at the location equivalent to 3 dB less than the peak gain on main lobe (in this case at 27 dB → Θ = 6°) For the same total transmitted power, a large peak Gain will correspond to a narrow beam width → desired Radar Meteorology M. D. Eastin Transmitted Power Relationship between Gain, Beam Width, Wavelength, and Antenna Size? • If the same antenna is used for both transmitting and receiving, then the antenna size is related to the Gain (since the effective beam width is) G 4Ae 2 10 cm (4) Where: Ae = effective antenna area λ = transmitting wavelength Thus: Large antennas → Large Gain → Small Beam Widths → Large Wavelengths 0.8 cm Desired: Small beam widths Small antennas Large gain Radar Meteorology M. D. Eastin Transmitted Power Problems Associated with Side Lobes: • Echo from the side lobe is interpreted as being from the main beam, but the return power is weak because the transmitted power was weak Horizontal “spreading” of weaker echo to sides of storm… Radar Meteorology M. D. Eastin Transmitted Power Problems Associated with Side Lobes: • Echo from the side lobe is interpreted as being from the main beam, but the return power is weak because the transmitted power was weak Vertical “spreading” of weaker echo to top of storm… Radar Meteorology M. D. Eastin Transmitted Power Method to Minimize Side Lobes: Beam Geometry • Use a parabolic antenna Outgoing Power Flux Density • Parabolic antennas allow for “tapered illumination” which minimizes 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 Radar Meteorology M. D. Eastin Backscatter Power Radar Cross Section: • 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) Target Backscatter Power Flux Density S (r ) back Sinc r (5) PROBLEM: We don’t measure Sback at r, we measure it at the radar Radar Transmitted Power Flux Density • For practical purposes, redefined as the power flux density received at the radar: Sr S inc 4r 2 Radar Meteorology (6) M. D. Eastin Backscatter Power Radar Cross Section: • In general, the radar cross section of a target depends on: 1. Target’s shape 2. Target’s size relative to the radar’s wavelength (more on this later …) 3. Complex dielectric constant and conductivity of the target (more on this later…) 4. Viewing aspect from the radar Radar Meteorology M. D. Eastin Power Received at Antenna Bringing it all together... • Recall from before: S inc GPt 4r 2 Sr S inc 4r 2 (3) Power flux density incident on target (6) Radar cross section (7) Power flux density of target backscatter at the antenna • Substituting (3) into (6): GPt Sr 16 2 r 4 • Accounting for the antenna area: Pr S r Ae Radar Meteorology AeGPt 16 2 r 4 (8) Ae is the effective antenna area (m2) Pr is the received power (W) M. D. Eastin Power Received at Antenna Bringing it all together… • Recall from before: AeGPt 16 2 r 4 Pr S r Ae G 4Ae 2 (8) Power flux density incident on target (6) Gain – Antenna size relationship • Substituting (6) into (8): 2G 2 Pt Pr 64 3 r 4 (9) Radar equation for a single isolated target (e.g. an airplane, ship, bird, one raindrop) • Let’s rearrange and examine this equation in more detail… Radar Meteorology M. D. Eastin Radar Equation for a Solitary Target 1 Pr 64 3 Constant r4 PG 2 2 t Radar Characteristics Target Characteristics • Written in terms of antenna effective area: 1 Pr 4 Constant Pt Ae2 2 Radar Characteristics r4 Target Characteristics • What do these equations tell us about radar returns from a single target? Radar Meteorology M. D. Eastin Distributed Targets Distributed Target: • A target consisting of many scattering elements, for example, the billions of raindrops that might be illuminated by a single radar pulse Contributing Region: • Volume containing all objects from which the scattered microwaves arrive back at the radar simultaneously • Spherical shell centered on the radar • Radial extent determined by the pulse duration • Angular extent determined by the antenna beam pattern Radar Meteorology M. D. Eastin Distributed Targets Single Pulse Volume: • Azimuthal coordinate (Θ) • Beam width in the azimuthal direction is rΘ, where Θ is the arc length between the half power points of the beam • Elevation coordinate (Φ) • Beam width in the elevation direction is rΦ, where Φ is the arc length between the half power points of the beam • Cross-sectional area of beam: r r 2 2 • Contributing volume length = half pulse length: c h 2 2 • Volume of contributing region for a single pulse: 2 2 h r r hr cr Vc 2 2 2 8 8 Radar Meteorology (10) M. D. Eastin Distributed Targets Single Pulse Volume: NEXRAD Radar • Pulse duration → τ = 1.57 μs • Angular circular beam width → 0.0162 radians • Range from radar → r = 100 km Vc cr 2 8 3.14 3.0 108 m s1 1.57106 s 105 m 0.0162 8 2 2 Vc 5.2 108 m3 • If the concentration of raindrops is the typical 1/m3, then the pulse volume contains: 520 million raindrops!! Radar Meteorology M. D. Eastin Distributed Targets Caveats to Consider: • The pulse volume is not a perfect cone • Recall the antenna beam (gain) pattern • About half the transmitted power fall outside the 3 dB cone • The gain function is not uniform with the cone – targets along the beam axis received more power than those off axis Radar Meteorology M. D. Eastin Distributed Targets Radar Cross-Section: Assumptions 1) The radial extent (h/2) of the contributing region is small compared to the range (r) so that the variation of Sinc across h/2 can be neglected (good assumption) 2) Sinc is considered uniform across the conical beam and zero outside – the spatial variation of the gain function can be ignored. (not good, but we are stuck with this one) 3) Scattering by other objects toward the contributing region must be small so that interference effects with the incident wave do not modify its amplitude (good for wavelengths > 3 cm) 4) Scattering or absorption of microwaves by objects between the radar and contributing region do not modify the amplitude of Sinc appreciably (good for wavelengths > 3 cm) Radar Meteorology M. D. Eastin Distributed Targets Radar Cross-Section: • Equal to the average radar cross for the random array of individual particles that comprise the target (e.g. average radar cross section of 520 million drops) j (11) j • Recall radar equation for a single target: 1 Pr 64 3 PG 2 2 t r4 • Radar equation for a distributed target: Pr Radar Meteorology 1 64 3 P G 2 t 2 j j r4 (12) M. D. Eastin Distributed Targets Radar Reflectivity (ηavg): • Since the radar cross-section is valid for all targets within the contributing volume, and that volume is non-uniform (i.e. its a function of range and beam shape), we need to account for this variability in each possible contributing volume: j Vc j Vc Vc avg (13) • Using (13) and the definition of the contributing volume (10), our radar equation for a distributed target becomes: c Pr 512 2 Radar Meteorology PG 2 t 2 avg 2 r (14) M. D. Eastin Radar Equation for Distributed Targets Accounting for Gain Function Shape: • Since the gain function maximizes along the beam axis, and decreases with angular distance from the axis, we can approximate this shape as a Gaussian function c Pr 512 2 avg 2 r PG 2 2 t • The equation above assumes uniform beam • A correction factor of 1/[2ln(2)] is needed for Gaussian-shaped beams c Pr 1024ln(2) 2 Constant Radar Meteorology PG 2 2 t Radar Characteristics avg 2 r Target Characteristics Radar equation for a distributed target (multiple birds) (multiple aircraft) (multiple raindrops) M. D. Eastin Distributed Weather Targets Modifying our Radar Equation for Weather Targets: • Since meteorologists are interested in weather targets, we can develop special forms of the distributed radar equations for typical collections of precipitation particles. Three tasks must be completed: 1) Find the radar cross section of a single precipitation particle 2) Find the total radar cross section for the entire contributing region 3) Obtain the average radar reflectivity from all particles in that region First Assumption: All particles are spheres! Small Raindrops Large raindrops Ice Crystals Gaupel / Hail Radar Meteorology Spheres Ellipsoids Variety of shapes Variety of shapes M. D. Eastin Distributed Weather Targets Second Assumption: All particles are sufficiently small compared to the wavelength of the transmitted radar pulse such that the back scatter can be described by Rayleigh Scattering Theory Types of scattering: Rayleigh Mie Optical How small? Why Raleigh scattering? • Radius less than λ/20 • Since the particle is much smaller than the variability associated with the radar pulse E-field (a sine wave), then we can assume the E-field across the particle will be uniform Radar Meteorology M. D. Eastin Distributed Weather Targets Impact of Radar Pulse on a Water Particle: • The radar pulse, and its associated E field, will induce an electric dipole within any homogeneous dielectric sphere (i.e. a water drop or ice sphere) • The induced dipole vector p 0 KD 3 Einc → Points in the same direction as the pulse’s E field → Magnitude is the product of the incident field and the polarization of the sphere: where: 2 ε0 = permittivity of free space K = dielectric constant for water/ice D = diameter of sphere Einc = amplitude of incident E field • The sphere then scatters that portion of the E field equivalent to the dipole magnitude • The backscatter received at the radar is: p Er 2 0r Radar Meteorology OR 2 KD 3 Einc Er 22 r M. D. Eastin Distributed Weather Targets Radar Cross Section of a Small Dielectric Sphere: • Recall the definition of radar backscatter: Sr S inc 4r 2 • The power flux densities and the E-field are related via: Sr Er2 2 Sinc Einc • Using the previous three equations, the radar cross section for a single sphere: 5 K 2 D6 4 Radar Meteorology Proportional to the sixth power of the diameter Proportional to the inverse fourth power of the radar wavelength M. D. Eastin Distributed Weather Targets What is the Dielectric Constant (K2)? • A measure of the scattering and absorption properties of a medium (water or ice) r 1 K r 2 where: r 1 0 Permittivity of medium Permittivity of vacuum Values of K2: WATER 0.930 (spheres) ICE 0.176 (spheres) 0.202 (snow flakes) Radar Meteorology M. D. Eastin Distributed Weather Targets Radar Cross Section of Multiple Dielectric Spheres: • Following the same methods as before: 5 K 2 D6 4 • For an array of particles, we compute the average radar cross section: 5K 2 j 4 D6j j j • We then determine the average radar reflectivity: avg Radar Meteorology j j Vc 6 D j 5K 2 j 4 Vc M. D. Eastin Distributed Weather Targets Radar Reflectivity Factor (Z) for Multiple Dielectric Spheres: • Most practitioners of radar use this quantity to characterize precipitation Z 6 D j j Thus… avg Vc 5K 2Z 4 • It is regularly expressed in logarithmic units Z dBZ 10 log 6 3 1m m / m Note the required units for Z to have a unitless dBZ and displayed on radar screens… Radar Meteorology M. D. Eastin Distributed Weather Targets Radar Equation: • We can then insert our radar reflectivity into our radar equation for a distributed target: avg 5K 2Z 4 c Pr 1024ln(2) 2 PG 2 2 t avg 2 r to get our desired radar equation for weather targets: c 3 PtG 2 K 2 Z 2 Pr 2 1024ln(2) r • Solving for Z: Pr r 2 1024ln(2) 2 2 Z 3 2 c PtG K Constant Radar Meteorology Radar Characteristics Target Characteristics M. D. Eastin Weather Radar Equation Review of Assumptions: 1) The precipitation particles are homogeneous dielectric spheres with diameters small compared to the radar wavelength 2) Particles are spread through the contributing region. If not, then the equation gives an average radar reflectivity factor for the contributing region 3) The reflectivity factor (Z) is uniform throughout the contributing region and constant over the period of time required to obtain the average value of the received power 4) All of the particles have the same dielectric constant. We assume they are all either water or ice spheres 5) The main lobe of the radar pulse is adequately described by a Gaussian function 6) Microwave attenuation over the distance between the radar and the target is negligible. 7) Multiple scattering is negligible. Since attenuation and multiple scattering are related, if one is true, both are true. 8) The incident and backscattered waves are linearly polarized, Radar Meteorology M. D. Eastin Weather Radar Equation Validity of the Rayleigh Approximation: Valid: λ = 10 cm λ = 3 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) 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) Invalid: Ice stones: > 2 cm (large hail) λ = 10 cm λ = 3 cm Radar Meteorology Ice crystals: > 0.5 cm (snowflakes) Ice stones: > 0.5 cm (hail and large graupel) M. D. Eastin Weather Radar Equation Equivalent Radar Reflectivity Factor (Ze): • If one or more of the assumptions built into the radar equation are not satisfied, then the reflectivity factor is re-named Pr r 2 1024ln(2) 2 2 Ze 3 2 c PtG K • In practice, one or more of the assumptions is almost always violated, and we regularly use Z and Ze interchangeably. Radar Meteorology M. D. Eastin Radar Equations Summary: • Basic Approach to Radar Equation Development • Solitary Target • Power incident on target • Power scattered back toward radar • Amount of power collected by the antenna • Distributed (Multiple) Targets • Distributed (Multiple) Weather Targets Radar Meteorology M. D. Eastin