Radar Detection: Swerling Models & Monte Carlo

What the Radar Equation Doesn't Tell You

The classic radar range equation gives one number: the range where received SNR reaches the detection threshold. That assumes the target's RCS is fixed, the atmosphere is transparent and the radar performs exactly to spec. Real targets don't cooperate.

Aircraft bank and yaw, changing their presented cross-section by 20 dB or more between looks. Rain attenuates the signal on the way out and on the way back. Transmitter power and receiver noise figure differ from unit to unit. What you need is detection probability as a function of range, with those effects included.

This walkthrough uses the Radar Detection Simulator on an S-band ground surveillance radar at 3 GHz.

Target Models: Choosing the Right Swerling Case

The five Swerling cases, from the 1950s, are still the standard because they map cleanly onto physical scattering:

Swerling 1 and 2 give the RCS an exponential distribution — many comparable scatterers interfering. Swerling 3 and 4 give it a chi-squared distribution with four degrees of freedom: one dominant scatterer plus smaller ones, so it fades less deeply. "Slow" means the RCS is constant across the pulses you integrate and changes between scans. "Fast" means it changes from pulse to pulse, so integration averages the fluctuation out.

The simulator averages detection probability over the RCS distribution exactly — $P_d = E_\sigma\left[P_d(\text{SNR}\cdot\sigma)\right]$ — rather than evaluating it at a single RCS percentile.

Setting Up the Nominal Case

The simulator computes per-pulse SNR from the monostatic radar equation

with $B = 1/\tau$. It integrates $N$ pulses non-coherently (Barton's $N^{0.8}$ gain, 8.0 dB for ten pulses), sets the threshold from Pfa with the chi-squared statistic, and maps SNR to detection probability with the noncentral chi-squared distribution averaged over the Swerling RCS density.

Result for Swerling 2:

How Much the Target Model Matters

At Pd = 0.5 the models differ by only 7 km. At Pd = 0.9 they differ by 30 km. A slowly fluctuating target that happens to present a small RCS for a whole dwell is simply missed, and integrating more pulses cannot help — every pulse sees the same fade. Swerling 1 needs 25 km less range than Swerling 2 to reach 90 % detection. If the radar is not frequency agile, design against Swerling 1.

Adding Rain: ITU-R P.838-3 Attenuation

Rain attenuation per kilometre follows

with the ITU-R P.838-3 coefficients. At 3 GHz, horizontal polarization, $k$ = 0.000139 and $\alpha$ = 1.232. The simulator applies the two-way loss, $2\gamma_R$, along the whole path.

At S-band, rain costs a few kilometres even in a downpour. That is why long-range air surveillance radars are built at L- and S-band.

Now take the same radar to 10 GHz (X-band). The shorter wavelength costs 10.5 dB of SNR through the $\lambda^2$ term, and P.838-3 gives $k$ = 0.0122, $\alpha$ = 1.257 — about 100 times the specific attenuation:

Heavy rain cuts the X-band radar's range by 60 %.

The uniform-path assumption is conservative for high targets: a real rain cell is a few kilometres across, and a target above the melting layer is partly outside it. For a low-altitude target inside a storm, it is close to reality.

Monte Carlo: Hardware Uncertainty

The target's RCS fluctuation is part of the physics and is already inside $P_d$. What remains uncertain is your hardware — you don't know exactly how the built radar will perform. The simulator varies transmit power (σ = 0.3 dB) and noise figure (σ = 0.5 dB) over 500,000 trials on a paid account and reports Pd curves at the 5th, 50th and 95th percentiles of that spread:

About ±5 km around the nominal. If your transmitter or receiver tolerances are wider than the tool's defaults, expect the band to widen roughly in proportion: range scales as the fourth root of SNR, so each dB of SNR is worth about 6 % of range.

Reading the ROC Curve

The ROC curve plots detection probability against false-alarm probability at the nominal Pd = 0.5 range (85.3 km):

Relaxing $P_{fa}$ from $10^{-6}$ to $10^{-4}$ raises $P_d$ from 0.49 to 0.75 at the same range. Whether that trade makes sense depends on the operation. Air traffic control needs very low false-alarm rates; a maritime search radar whose operator correlates contacts over several scans may accept $10^{-4}$ happily.

What This Simulation Won't Tell You

The simulator models thermal-noise detection, non-coherent integration, the five Swerling fluctuation models, P.838-3 rain loss and hardware uncertainty. It does not model clutter (ground, sea, weather or chaff), which can dominate thermal noise by 30 dB or more for low-altitude targets, nor jamming, multipath lobing, or antenna scan loss away from boresight. For a point target in free space it gives the probability framework the deterministic range equation lacks.

Updated 2026-09-19. An earlier version of this article quoted detection ranges, rain attenuation (ten times too high at 3 GHz) and Monte Carlo inputs that did not match the simulator. Every number above now comes from the tool. Radar Detection Simulator