Radar Detection Probability: Swerling Models and Monte Carlo Uncertainty Analysis

What the Radar Equation Doesn't Tell You

The classic radar range equation gives you a single number: the range at which received SNR equals your detection threshold. It assumes a point target with fixed RCS, no atmospheric losses, and perfect system parameters. Real radar targets don't work this way.

Aircraft flutter, ships roll, precipitation scatters — target radar cross-section fluctuates from pulse to pulse or scan to scan. Rain adds 0.01–20 dB/km of two-way path loss depending on frequency and rain rate. Your transmit power varies ±1 dB from unit to unit and ±2 dB with temperature. The radar range equation gives you a snapshot; the detection simulator gives you a probability distribution over that snapshot.

This walkthrough uses the Radar Detection Simulator to analyze a ground-based surveillance radar operating at 3 GHz.

Target Models: Choosing the Right Swerling Case

Pefore running a simulation, you need to pick a target fluctuation model. The five Swerling cases cover the range from optimistic to realistic:

For a fighter-sized aircraft at 3 GHz with pulse-to-pulse integration, Swerling 2 is the standard choice. Swerling 1 is more pessimistic (slow fluctuation makes integration less effective) and produces lower Pd at the same SNR — use it when you need a conservative link margin.

Setting Up the Nominal Case

Enter the following parameters for a 3 GHz ground surveillance radar:

The simulator computes SNR at each range bin using the Friis radar equation, then maps SNR to Pd using the Marcum Q-function (Swerling 0) or the appropriate noncentral chi-squared CDF for Swerling 1–4. Non-coherent integration of N pulses improves SNR by approximately N^0.8 for Swerling fluctuating targets.

With these inputs the nominal detection range (Pd = 0.5) comes out around 180 km. The 90% detection range is closer to 120 km — the range where nine out of ten scan opportunities will detect the target.

Adding Rain: ITU-R P.838 Attenuation

Now enable rain attenuation and set rain rate to 16 mm/hr (moderate rain, ITU-R climate zone K). The simulator applies the P.838 specific attenuation model:

where k and α are frequency-dependent coefficients. At 3 GHz with horizontal polarization, k ≈ 0.00155 and α ≈ 1.265, giving γ_R ≈ 0.044 dB/km at 16 mm/hr. Over a 180 km two-way path that's 16 dB of additional loss — enough to cut the detection range to about 120 km for the nominal case.

The rain region is limited to the first 4 km in altitude (the bright band), which the simulator handles via an effective path length reduction. Heavier rain (50 mm/hr — tropical thunderstorm) produces γ_R ≈ 0.21 dB/km and reduces nominal detection range below 90 km.

Monte Carlo: Quantifying System Uncertainty

Nominal detection range is the median — half of all manufactured radar systems will perform worse. Enable Monte Carlo with 50,000 trials and the following tolerances:

The Monte Carlo result shows that the 10th-percentile detection range (worst 10% of system-plus-environment combinations) is 95 km — 25% shorter than the nominal. The 90th-percentile (best 10%) reaches 155 km. This spread represents real manufacturing variance, seasonal noise figure drift, and target aspect angle variation.

The most influential parameter is target RCS, which drives nearly 60% of the detection range variance in the sensitivity breakdown. This is expected for Swerling 2 targets: RCS fluctuates pulse-to-pulse with a Rayleigh distribution, and the tails of that distribution dominate Pd at moderate SNR. The implication is that investing in higher transmit power or better antenna gain has diminishing returns if you haven't accounted for target aspect angle variance.

Reading the ROC Curve

The Receiver Operating Characteristic (ROC) curve plots Pd against Pfa for a fixed range. Use it to answer: "if I relax my false alarm rate from 10⁻⁶ to 10⁻⁴, how much do I gain in detection probability at 150 km?"

At 150 km with the nominal parameters and no rain, the ROC shows Pd rising from 0.41 at Pfa=10⁻⁶ to 0.68 at Pfa=10⁻⁴. That's a 27 percentage point gain in Pd for two orders of magnitude more false alarms — a tradeoff that depends entirely on the operational context. For air traffic control, Pfa=10⁻⁶ is mandatory. For a maritime search radar with a human operator screening contacts, Pfa=10⁻⁴ may be acceptable.

What This Simulation Won't Tell You

The simulator models thermal noise detection, range-Doppler processing gain (via non-coherent integration), rain attenuation, and target RCS fluctuation. It does not model clutter (ground, sea, chaff), ECM/jamming, multipath, or antenna scanning loss. For a full radar system analysis, those effects need separate models — but for link budget validation and detection range sensitivity analysis, this simulation gives you the essential probability framework.

Radar Detection Simulator