Surveillance Radar Lab Blender 5.2

A generic air-traffic and weather radar on a hill: the range equation, pulse timing, Doppler and MTI, the radar horizon, terrain masking and ground clutter, with airliners and a rain front as targets. Drag to orbit, wheel or pinch to zoom. Terrain is drawn with the height exaggerated.

Radar scope (PPI)

Display Range Fluctuating RCS (Swerling I)

Green-yellow-red: rain echo. Amber-brown: ground clutter. Circles: airliner plots that were detected on this sweep (Bernoulli draw with the computed probability of detection). A star means a second-trip (range-folded) echo.

Presets

Transmitter and receiver

Antenna and site

Signal processing

Weather, clutter and atmosphere

Radar numbers

Airliners and what the radar makes of them

Selected Marker size

RCS is the mean value of the aircraft class times an aspect factor 0.25 + 1.5 sin²(aspect angle): weak nose-on, strong broadside. The verdict is computed from the full chain: terrain line of sight, elevation pattern, rain path loss, MTI response, ground-clutter and rain interference.

Echo against range (A-scope)

Power after the receiver at the current beam azimuth, relative to thermal noise. Spikes are airliners, the broad hump is rain, the jagged low part is ground clutter (after the MTI improvement factor if MTI is on). The dashed line is the detection threshold from the Albersheim formula; the dotted vertical line is the unambiguous range.

Terrain section, horizon and shadow

Follow antenna azimuthAltitude window

Altitude against ground range along one azimuth, with the 4/3-earth bulge. Brown: terrain. Red: altitudes the radar cannot see because a ridge or the horizon blocks the straight line. Green lines: centre and half-power edges of the beam. Dots: airliners within a few degrees of this azimuth.

Range equation: SNR against range

Single-pulse SNR for the selected airliner (current RCS) falls 40 dB per decade of range (R⁴). Dashed: the Albersheim threshold for the current Pd, Pfa and pulses per scan. Vertical lines: free-space range (green) and the radar-horizon distance for this altitude (amber).

Doppler and MTI: blind speeds

Power response of the single delay-line canceller against radial speed, f_d = 2v/λ. Zero at v = nλPRF/2 (blind speeds) and at v = 0 (clutter notch). Staggering the PRF fills the first blind speeds. Dots: radial speed of each airliner now.

Weather sensitivity

Minimum detectable reflectivity (SNR = 3 dB) against range from the weather radar equation, with the actual reflectivity along the current beam azimuth (green).

Range resolution

Experiment: how far can a bigger target be seen?

Question: if the target's radar cross-section grows by a factor of 10, how much further is it detected? The radar equation says R ∝ σ^1/4, so only 1.78 times. The test places a point target on the beam axis in free space and moves it outward until the single-pulse SNR drops below the detection threshold, for RCS from 0.01 to 1000 m², then fits the slope on a log-log plot.

Add rain and gas loss

Press the button to sweep the target RCS.

Detection range against RCS (log-log). Dots: measured by moving the target. Line: closed-form R_max = [P G² λ² σ / ((4π)³ S_min L)]^1/4. Dotted: slope exactly 1/4.

View

Independent numerical checks

The science

The radar range equation

A pulse of peak power P leaves an antenna of gain G, so the power density at range R is P G / (4πR²). A target of radar cross-section σ (the area of a perfectly isotropic scatterer that would return the same echo) re-radiates, and the receiving antenna of effective area A_e = Gλ²/(4π) collects P_r = P G² λ² σ / ((4π)³ R⁴ L). The two spreading losses multiply, which is why the echo falls with the fourth power of range. Thermal noise in the matched bandwidth B ≈ 1/τ is N = k T₀ B F; detection needs P_r ≥ N · SNR_req. Solving for R gives R_max = [P G² λ² σ / ((4π)³ S_min L)]^1/4, so a ten times larger target is seen only 10^1/4 = 1.78 times further, and doubling the power buys 19 % in range. L collects system losses (plumbing, beam shape, processing: 12 dB here), radome, rain and gas.

Detection threshold

The required single-pulse SNR comes from Albersheim's approximation for Pd and Pfa with N pulses integrated non-coherently: SNR_dB = −5 log₁₀N + [6.2 + 4.54/√(N+0.44)] log₁₀(A + 0.12AB + 1.7B), A = ln(0.62/Pfa), B = ln(Pd/(1−Pd)). N is the number of pulses that hit the target per scan, PRF · θ_az / (6 · rpm). Real targets fluctuate: for a Swerling I target, Pd = Pfa^1/(1+SNR); this page applies it with an N^0.8 integration efficiency, an approximation.

Pulse timing, resolution and ambiguity

An echo returns after t = 2R/c. Pulses are sent every 1/PRF, so an echo from beyond R_u = c / (2 PRF) arrives after the next pulse and is plotted at R mod R_u: a second-trip (range-folded) ghost. Two scatterers merge unless they are further apart than ΔR = cτ/2. Beamwidth is set by the aperture: θ ≈ 1.2 λ/D for a circular dish, 1.1 λ/W for a rectangular reflector (tapered illumination); gain G = η(πD/λ)² with aperture efficiency η = 0.6. The elevation pattern is either a pencil beam or a cosecant-squared fan (G ∝ csc² el above the pencil, 3 dB lower peak) which holds constant-altitude coverage out to 35°.

Doppler, MTI and blind speeds

A target moving at radial speed v shifts the echo by f_d = 2v/λ. Pulse to pulse the phase advances 4πvT/λ; a single delay-line canceller subtracts consecutive pulses, leaving a power response sin²(π f_d / PRF): zero for stationary clutter, and also zero at v_b = n λ PRF / 2, the blind speeds. The Doppler velocity measured by a pulse-pair radar is unambiguous only up to ±λ PRF/4, so storm winds fold. Staggering two PRIs (4:5) is modelled by averaging the two responses.

Radar horizon, terrain masking, clutter

The atmosphere bends radar rays slightly downward; replacing the earth radius by 4/3 R_e gives straight rays and the horizon d = √(2 k R_e h₁) + √(2 k R_e h₂) ≈ 4.12 (√h₁ + √h₂) km (h in metres). On a flat map the ray height is h_r + m s + s²/(2kR_e); the page marches that line over the real height map and reports the lowest altitude visible at each range and azimuth. Ground clutter per resolution cell is σ_c = σ₀ R θ_az L_g with σ₀ = γ sin ψ (constant-gamma model: −30 dB flat farmland to −11 dB mountains, −10 dB towns) and L_g the smaller of the pulse and beam footprints; MTI divides it by an improvement factor (28 dB typical).

Weather radar and rain attenuation

Rain is a distributed target: the received power is P_r = π³ c P τ G² θφ |K|² Z / (1024 ln2 λ² R²) (Probert-Jones), with Z in mm&sup6;/m³, Z = 200 R^1.6 (Marshall-Palmer) for rain rate R in mm/h. Falling only as R−², rain is seen much further than a point target. Rain also absorbs the beam, two-way, at k R^α dB/km one way (ITU-R P.838-like coefficients: 3 GHz 0.0007 R^1.09, 9.4 GHz 0.0097 R^1.28 dB/km), which is why an X-band radar loses everything behind a squall line and an S-band radar does not.

Educational model. The height map, rain cells, airliners and station are generic inventions; RCS values, losses and gammas are round textbook numbers, not those of any real installation or aircraft. Ground clutter, speckle and the Swerling treatment are simplified; multipath, ducting, sidelobe clutter, bird returns, and signal-processing details such as pulse compression are not modelled. The station is drawn about 25 times larger than real so that it is visible on a 240 km map.