M7. Weather & Atmospheric Radar

Rain, snow, hail, cloud, and insects all scatter microwaves. Weather radar exploits Rayleigh scattering to measure precipitation, Doppler to detect wind shear and tornado signatures, and dual polarization to discriminate hydrometeor types. From NEXRAD saving lives in tornado alley to cloud profilers at 94 GHz, atmospheric radar has its own rich physics.

1. Rayleigh Scattering from Hydrometeors

For a spherical particle of diameter $D$ much smaller than the wavelength ($D\ll\lambda$), the backscatter cross section is

$$\boxed{\;\sigma_b = \frac{\pi^5}{\lambda^4}\,|K|^2\,D^6\;},\qquad K = \frac{\varepsilon_r - 1}{\varepsilon_r + 2}.$$

The factor $|K|^2$ depends on the dielectric constant: for liquid water at S-band,$|K|^2 \approx 0.93$; for ice, 0.18. The $D^6/\lambda^4$ scaling has two profound consequences:

Large drops dominate: a single 5 mm drop returns as much power as 15000 drops of 1 mm.
Short wavelengths see more power but attenuate faster — X-band is ideal for detail, S-band for penetration.

2. Reflectivity Factor Z and dBZ

Integrating $\sigma_b$ over the drop-size distribution (DSD) $N(D)$:

$$\eta = \int \sigma_b(D)\,N(D)\,dD = \frac{\pi^5|K|^2}{\lambda^4}\int N(D)\,D^6\,dD = \frac{\pi^5 |K|^2}{\lambda^4}\,Z.$$

The reflectivity factor $Z \equiv \int N(D)D^6\,dD$ [mm\u2076/m\u00B3] is an intrinsic property of the scatterers, independent of wavelength. Expressed in decibels:

$$\text{dBZ} = 10\log_{10}\left(\frac{Z}{1\,\text{mm}^6/\text{m}^3}\right).$$

Interpretation: <20 dBZ light drizzle, 30-40 dBZ moderate rain, 50+ dBZ heavy convection, 60+ dBZ likely hail.

3. The Z-R Relation

Rain rate $R$ in mm/hr is the first moment of the product of $N(D)$, drop volume, and terminal velocity. Empirically $Z$ and $R$ are related by a power law:

$$\boxed{\;Z = a\,R^{b}\;},\qquad \text{Marshall-Palmer: } a=200,\;b=1.6.$$

Other common choices:

Regime	a	b
Marshall-Palmer (stratiform)	200	1.6
WSR-88D default	300	1.4
Tropical convective	250	1.2
Snow (dry)	2000	2.0

Z-R estimation of rainfall has a factor-of-2 uncertainty without correction; dual-polarization variables below reduce this to 15-20%.

4. Drop-Size Distributions

Marshall and Palmer (1948) proposed an exponential DSD:

$$N(D) = N_0\,e^{-\Lambda D},\qquad N_0 = 8\times 10^6\,\text{m}^{-3}\text{mm}^{-1},\qquad \Lambda = 4.1\,R^{-0.21}\,\text{mm}^{-1}.$$

More realistic: Gamma DSD (Ulbrich 1983) with shape parameter $\mu$:

$$N(D) = N_0\,D^{\mu}\,e^{-\Lambda D}.$$

Three-parameter Normalized-Gamma (Testud 2001) decouples water content, median drop diameter, and shape — enabling retrievals from 2+ radar variables.

5. Doppler Weather Radar

Each resolution cell contains $10^6$+ scatterers; the received signal is Gaussian with power spectrum centered at the mean radial velocity. Key moments:

Zeroth moment: total power = reflectivity Z.
First moment: mean velocity $\bar v_r$ = Doppler centroid.
Second moment: spectrum width $\sigma_v$ = turbulence + wind shear + antenna rotation.

Pulse-pair processing (PPP) estimates $\bar v_r$ and $\sigma_v$ from lag-1 autocorrelation, avoiding expensive FFTs:

$$\bar v_r = -\frac{\lambda}{4\pi\,\text{PRI}}\arg R(1),\qquad \sigma_v^2 = \frac{\lambda^2}{8\pi^2\text{PRI}^2}\ln\frac{|R(0)|}{|R(1)|}.$$

Velocity folding, range-velocity ambiguity (Doppler dilemma), and ground-clutter filtering are the major operational challenges.

6. Dual-Polarization Radar

Transmitting and receiving both H and V polarizations yields a suite of variables:

Variable	Definition	Physical meaning
Z_H	\|S_HH\|\u00B2	Reflectivity, horiz pol
Z_DR	10 log(Z_H/Z_V)	Differential reflectivity: drop shape
\u03C1_HV	\|<S_HH S_VV*>\|/\u221A(<\|S_HH\|\u00B2><\|S_VV\|\u00B2>)	Co-pol correlation: mixed hydrometeor
\u03A6_DP	arg(<S_VV S_HH*>)	Differential phase: path-integrated
K_DP	d\u03A6_DP/dr /2	Specific differential phase: rainfall rate
LDR	10 log(\|S_HV\|\u00B2/\|S_HH\|\u00B2)	Linear depolarization ratio: non-Rayleigh

Large oblate raindrops have $Z_{DR} > 0$; tumbling hail has $Z_{DR}\approx 0$; ice crystals may have $Z_{DR} < 0$. Uniform spherical scatterers give $\rho_{HV} \to 1$; mixed-phase precipitation drops it to 0.95; biological scatterers even lower.

R(K_DP) rainfall estimators use the phase variable (immune to attenuation, absolute calibration) and give 20-30% rainfall accuracy in heavy rain — the standard for operational QPE since the NEXRAD dual-pol upgrade (2013).

7. Hydrometeor Classification

Fuzzy-logic classifiers map the tuple $(Z_H, Z_{DR}, \rho_{HV}, K_{DP}, T, H)$ to classes: drizzle, light rain, heavy rain, big drops, wet snow, dry snow, ice crystals, graupel, hail, biological, ground clutter. Park et al. (2009) is the standard WSR-88D classifier. Neural networks trained on disdrometer + radar data now match or exceed fuzzy logic.

8. Weather-Radar Equation

For distributed targets filling the beam, the radar equation is modified: the illuminated volume scales as $R^2$ (beam area) times $c\tau/2$ (pulse length):

$$P_r = \frac{P_t G^2 \lambda^2}{(4\pi)^3 R^2}\cdot \eta \cdot \frac{\pi\theta_{3\text{dB}}^2(c\tau/2)}{8\ln 2}\cdot L.$$

Note the $R^{-2}$ dependence (not $R^{-4}$) — distributed targets see a larger illuminated volume at long range. Substituting $\eta = (\pi^5|K|^2/\lambda^4)Z$gives the $|K|^2 Z$ detectability at the receiver.

PPI Display

9. NEXRAD and Operational Weather Radar Networks

NEXRAD (Next Generation Weather Radar) / WSR-88D is the US network of 160 S-band Doppler radars, operational since 1988 and dual-pol upgraded 2011-2013. Specifications:

Frequency: 2.7-3.0 GHz (S-band)
Peak power: 750 kW klystron
Antenna: 8.5 m diameter parabola, 0.95\u00B0 beamwidth, 45 dB gain
Range: 460 km reflectivity, 230 km velocity
Volume scan: 14 elevations in 5-10 min (VCP)
dBZ accuracy: \u00B11 dB after calibration

Equivalent European networks: OPERA (composite of 200+ C-band radars). Japan AMeDAS operates C- and X-band. China has a ~250-radar S/C-band network. All data feeds numerical weather prediction via ECMWF, HRRR, JMA MSM assimilation systems.

10. Non-Rayleigh Scattering: Mie and Resonance

For larger drops $D\sim\lambda$, the Rayleigh approximation breaks down. Mie theory gives oscillating backscatter cross section; for 5 mm drops at X-band ($kD\sim 1$), Mie exceeds Rayleigh by factors of 3-10. This matters for hail, which can be centimeters in size and exhibits strong resonance features, especially at higher frequencies.

The T-matrix method extends Mie to non-spherical particles (oblate raindrops, hail spheroids, needle crystals) and is the workhorse of modern weather-radar simulators.

11. Profilers and Cloud Radars

Specialized atmospheric radars:

Wind profilers (VHF/UHF, 50-1000 MHz): measure vertical wind from Bragg scattering off refractive-index turbulence. Range to 15 km tropo, stratosphere.
Cloud radars (Ka, W band, 35 and 94 GHz): detect non-precipitating cloud droplets (sub-mm) via Rayleigh scattering. Used at ARM sites, NASA CloudSat, EarthCARE.
Precipitation radars from space: TRMM (1997-2015), GPM (2014-) provide 3-D global precipitation. Both use Ku + Ka dual-frequency.
Mesosphere-Stratosphere-Troposphere (MST): VHF arrays observing 0-100 km refractivity layers.

12. Rain Attenuation and Correction

At C-band and above, rain attenuates the radar beam significantly. Specific attenuation$A_H$ is related to differential phase $K_{DP}$ approximately linearly (Bringi & Chandrasekar):

$$A_H \approx \alpha\,K_{DP},\quad \alpha\approx 0.08\text{ (C-band)},\;0.25\text{ (X-band)}.$$

This allows self-consistent attenuation correction directly from phase measurements. The ZPHI algorithm (Testud 2000) is operational on European C-band radars.

13. Ground Clutter Filtering

Weather returns are typically 30-60 dB weaker than ground clutter at low elevation angles. IIR notch filters (Elliptic) in slow-time remove zero-Doppler clutter; the GMAP (Gaussian Model Adaptive Processing) estimates and removes a narrow clutter spectrum while preserving low-velocity weather signal. Dual-pol adds \u03C1_HV as a clutter indicator (close to 0.5-0.8 for ground clutter).

Simulation: Rayleigh Scattering, DSDs, Z-R Relations

The Python simulation computes Rayleigh backscatter cross sections for drop diameters 0.1-10 mm at S/C/X bands, plots the Marshall-Palmer drop-size distribution for rain rates 1-100 mm/hr, integrates the DSD to obtain the Z-R relation, and produces the dBZ-vs-R curve annotated with standard precipitation intensities.

Python

script.py96 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ---- Rayleigh scattering from hydrometeors ----
# sigma_b = (pi^5 |K|^2 / lam^4) * D^6
lam_sband = 0.10    # S-band 10 cm
lam_cband = 0.053
lam_xband = 0.03
K_water_sq = 0.93   # dielectric factor for water at S-band

D = np.logspace(-4, -2, 400)  # drop diameter 0.1 mm to 10 mm
def sigma_back(D, lam, K2):
    return np.pi**5 * K2 / lam**4 * D**6

sig_s = sigma_back(D, lam_sband, 0.93)
sig_c = sigma_back(D, lam_cband, 0.93)
sig_x = sigma_back(D, lam_xband, 0.93)

# ---- Marshall-Palmer DSD ----
# N(D) = N0 exp(-Lambda D), Lambda = 4.1 R^-0.21
def ND(D, R):
    N0 = 8e6  # m^-3 mm^-1
    lam_mp = 4.1 * R**(-0.21)  # mm^-1
    return N0 * np.exp(-lam_mp * D*1000)  # convert D to mm

# ---- Z (reflectivity factor) ----
# Z = integral N(D) D^6 dD [mm^6/m^3]
def Z_from_R(R):
    Dmm = np.linspace(0.1, 8, 400)
    N = ND(Dmm/1000, R)
    return np.trapezoid(N * Dmm**6, Dmm)

R_vals = np.linspace(0.1, 100, 200)
Z_vals = np.array([Z_from_R(R) for R in R_vals])
dBZ = 10*np.log10(Z_vals)

# Marshall-Palmer: Z = 200 R^1.6
Z_mp = 200 * R_vals**1.6

# ---- Plot ----
fig, axes = plt.subplots(2, 2, figsize=(14, 10))
fig.patch.set_facecolor('#0f172a')
for ax in axes.flat:
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1')
    for sp in ax.spines.values(): sp.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.4)

axes[0,0].loglog(D*1000, sig_s, color='#f87171', linewidth=1.8, label='S-band (10 cm)')
axes[0,0].loglog(D*1000, sig_c, color='#fbbf24', linewidth=1.8, label='C-band (5.3 cm)')
axes[0,0].loglog(D*1000, sig_x, color='#60a5fa', linewidth=1.8, label='X-band (3 cm)')
axes[0,0].set_xlabel('Drop diameter [mm]', color='#e2e8f0')
axes[0,0].set_ylabel(r'Backscatter $\sigma_b$ [m²]', color='#e2e8f0')
axes[0,0].set_title('Rayleigh scattering: σ ∝ D⁶/λ⁴', color='#fecaca', fontsize=13)
axes[0,0].legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

# DSD for a few rain rates
Dmm = np.linspace(0.1, 8, 400)
for R, col in zip([1, 5, 25, 100], ['#34d399','#fbbf24','#fb923c','#f87171']):
    axes[0,1].semilogy(Dmm, ND(Dmm/1000, R), color=col, linewidth=1.7, label=f'R={R} mm/hr')
axes[0,1].set_xlabel('Diameter [mm]', color='#e2e8f0')
axes[0,1].set_ylabel('N(D) [m⁻³ mm⁻¹]', color='#e2e8f0')
axes[0,1].set_title('Marshall-Palmer Drop Size Distribution', color='#fecaca', fontsize=13)
axes[0,1].legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

# Z-R relation
axes[1,0].loglog(R_vals, Z_vals, color='#a78bfa', linewidth=2, label='Integrated DSD')
axes[1,0].loglog(R_vals, Z_mp, '--', color='#fbbf24', linewidth=1.8, label='Z=200R^1.6 (MP)')
axes[1,0].set_xlabel('Rain rate R [mm/hr]', color='#e2e8f0')
axes[1,0].set_ylabel('Z [mm⁶/m³]', color='#e2e8f0')
axes[1,0].set_title('Z-R Relation', color='#fecaca', fontsize=13)
axes[1,0].legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

# dBZ vs R
axes[1,1].semilogx(R_vals, dBZ, color='#34d399', linewidth=2)
axes[1,1].axhline(20, color='#60a5fa', linestyle='--', alpha=0.5); axes[1,1].text(0.15, 22, 'Light', color='#60a5fa')
axes[1,1].axhline(40, color='#fbbf24', linestyle='--', alpha=0.5); axes[1,1].text(0.15, 42, 'Moderate', color='#fbbf24')
axes[1,1].axhline(55, color='#f87171', linestyle='--', alpha=0.5); axes[1,1].text(0.15, 57, 'Heavy', color='#f87171')
axes[1,1].axhline(65, color='#ef4444', linestyle='--', alpha=0.5); axes[1,1].text(0.15, 67, 'Hail', color='#ef4444')
axes[1,1].set_xlabel('Rain rate [mm/hr]', color='#e2e8f0')
axes[1,1].set_ylabel('Reflectivity [dBZ]', color='#e2e8f0')
axes[1,1].set_title('dBZ vs Rain Rate (precipitation intensity)', color='#fecaca', fontsize=13)

plt.tight_layout()
plt.savefig('output.png', dpi=130, bbox_inches='tight', facecolor='#0f172a')

print(f"Rayleigh backscatter of 1 mm drop at S-band: {sigma_back(0.001, lam_sband, 0.93):.2e} m^2")
print(f"Rayleigh backscatter of 1 mm drop at X-band: {sigma_back(0.001, lam_xband, 0.93):.2e} m^2")
print(f"Ratio (S/X) = {sigma_back(0.001, lam_sband, 0.93)/sigma_back(0.001, lam_xband, 0.93):.2f} (= (lam_x/lam_s)^-4 = {(lam_sband/lam_xband)**4:.2f})")
print()
for R in [1, 5, 25, 50, 100]:
    Z_val = Z_from_R(R)
    print(f"R = {R:>4} mm/hr  →  Z = {Z_val:8.0f} mm^6/m^3  =  {10*np.log10(Z_val):5.1f} dBZ")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

14. Radar Data Assimilation

Numerical Weather Prediction (NWP) assimilates radar reflectivity and radial velocity via 3D-Var, 4D-Var, and Ensemble Kalman Filter methods. The MRMS (Multi-Radar Multi-Sensor) system in the US produces 1 km composite reflectivity every 2 minutes. ECMWF assimilates global radar data into its IFS; operational mesoscale models (HRRR, AROME) run 1 km convection-allowing forecasts anchored on radar observations.

15. Severe Weather Signatures

Operational forecasters look for characteristic radar signatures:

Hook echo: reflectivity hook on the rear flank of a supercell — tornado warning indicator.
Velocity couplet: adjacent inbound-outbound pixels (±30 m/s) = mesocyclone.
Tornadic Debris Signature (TDS): low Z_DR, low \u03C1_HV in the debris cloud — confirmed tornado on ground.
Bow echo: bow-shaped reflectivity = damaging straight-line winds.
Bright band: enhanced Z around 0\u00B0C isotherm = melting layer.

16. Velocity-Azimuth Display (VAD) Wind Retrieval

A single Doppler radar rotates its antenna and measures radial velocity as a function of azimuth at a fixed elevation. Assuming horizontally uniform wind $(u, v)$, the radial velocity is

$$v_r(\phi) = u\sin\phi\cos\theta + v\cos\phi\cos\theta + w\sin\theta.$$

Fitting a sinusoid of azimuth recovers wind speed and direction at each height (elevation sweep). VAD provides the vertical wind profile assimilated into weather models every 6 minutes.

17. Bragg Scattering and Clear-Air Radar

Even cloud-free air scatters radar: refractive-index fluctuations from turbulence have a spatial spectrum $\Phi(k)\propto k^{-11/3}$ (Kolmogorov inertial subrange). A radar at wavenumber $k_r = 4\pi/\lambda$ is sensitive to the Bragg wavelength $\lambda/2$. At VHF (50 MHz, $\lambda = 6$ m), the Bragg scale is 3 m — deep in the inertial subrange — and returns are strong enough to profile winds to 100+ km altitude. This is the basis of MST radars (Mesosphere-Stratosphere-Troposphere).

18. Phased-Array Weather Radar

NOAA's PAR (Phased Array Radar) prototype achieves full-volume scans in ~1 minute vs 5-10 min for mechanically scanned NEXRAD. This enables observation of rapid-evolving tornado genesis, microbursts, and hail core development. Dual-pol phased arrays remain an active research problem because H and V patterns match poorly off-broadside. Expected deployment: late 2020s to replace aging NEXRAD klystrons.

19. Dual-Wavelength Retrievals

Observing the same hydrometeors at two frequencies gives the Dual-Frequency Ratio$\text{DFR} = Z_{Ku} - Z_{Ka}$ [dB], which is directly related to median drop diameter through Mie theory. GPM's dual-frequency precipitation radar (Ku + Ka, 2014-) is the flagship instrument. Space-based lidar-radar combinations (CALIPSO + CloudSat) additionally separate cloud water from ice.

20. Hail Detection

Hail stones of $D \ge 2$ cm are hazardous. Radar indicators: Z > 55 dBZ combined with Z_DR near 0 (tumbling), and high VIL (Vertically Integrated Liquid). Three-body scattering signatures (“flare echoes”) appear behind strong hail cores and confirm their presence.

21. Airborne and Shipborne Weather Radar

Commercial aircraft carry X-band nose radars to detect storms ahead and enable rerouting; modern systems integrate turbulence, windshear, and volcanic ash detection. Marine radars (X-band, 3 cm, 50 kW) operate under SOLAS for collision avoidance but also resolve 1-10 km-scale storm cells. Research radars on the NOAA P-3 Orions penetrate hurricanes directly, collecting eyewall and rainband Doppler data for storm-surge and intensity forecasting.

References

Doviak, R.J. & Zrnic, D.S. — Doppler Radar and Weather Observations, 2nd ed., Dover (2006). The classic.
Bringi, V.N. & Chandrasekar, V. — Polarimetric Doppler Weather Radar, Cambridge (2001).
Marshall, J.S. & Palmer, W.M. — “The distribution of raindrops with size”, J. Meteorol., 5, 165 (1948).
Ulbrich, C.W. — “Natural variations in the analytical form of the raindrop size distribution”, JCAM, 22, 1764 (1983).
Park, H.S., Ryzhkov, A.V., Zrnic, D.S., Kim, K.-E. — “The hydrometeor classification algorithm for the polarimetric WSR-88D”, Wea. Forecasting, 24, 730 (2009).
Testud, J. et al. — “The concept of 'normalized' distribution to describe raindrop spectra”, JAMC, 40, 1118 (2001).
Rauber, R.M. & Nesbitt, S.L. — Radar Meteorology: A First Course, Wiley (2018).
Skolnik, M.I. — Introduction to Radar Systems, ch. 10 (weather radar).

Cross-References

Atmospheric Science Climatology / Meteorology M3: Doppler

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