How do scan-to-scan correlation and Swerling fluctuation affect the cumulative calculation?

The simple product rule Pc = 1 minus the product of (1 minus Pd) assumes the detection decisions are statistically independent from scan to scan. This holds well for Swerling I and III targets that decorrelate between scans (slow fluctuation), where each scan sees an independent RCS sample. For Swerling II and IV targets that decorrelate pulse to pulse but where the mean is stable, scan independence still applies at the decision level. When the target RCS is correlated across scans, for example a steady Swerling 0 (Marcum) target with stable aspect, the scans are not fully independent and the true Pc is lower than the product rule predicts. Designers either use measured blip-scan data or apply a correlation factor to avoid overestimating cumulative performance.

Radar & Defense

Cumulative Detection Probability

Q: How does cumulative detection probability differ from single-scan probability of detection?

Single-scan probability of detection (Pd) is the chance of declaring a target on one specific look, given the instantaneous signal-to-noise ratio and a fixed false-alarm threshold. Cumulative detection probability (Pc) is the chance of detecting the target on at least one of N successive scans. For statistically independent scans, Pc = 1 minus the product of (1 minus Pd_i) over all i scans. Because the target closes in range over a search, the SNR rises roughly as 1/R^4, so the per-scan Pd climbs from near zero at long range to near one at short range, and Pc accumulates accordingly. A target with only 0.3 single-scan Pd reaches about 0.97 cumulative probability after 10 independent scans at that same level.

Q: What is cumulative detection range and why is it longer than single-scan range?

Cumulative detection range is the range at which the cumulative probability of detection first reaches a specified value, typically 0.9, as the target approaches. It is longer than the single-scan 0.9 detection range because each additional scan adds another independent chance to detect. The blip-scan ratio (the fraction of scans producing a detection at a given range) determines how quickly Pc builds. A radar with a 10 second frame time that needs Pc = 0.9 against an inbound Mach 1 target gains several extra scans during the closing geometry, extending the effective detection range by roughly 15 to 30 percent over the single-look range.

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Across a search, a radar gets repeated looks at an inbound target, and the chance of declaring it on at least one of those looks is the cumulative detection probability (P_c). Where the single-scan detection probability describes one look at a fixed signal-to-noise ratio, P_c combines the per-scan odds over N successive scans as the target closes and the SNR rises with the radar range equation. For statistically independent scans, P_c = 1 − ∏(1 − P_d,i), so even modest per-scan values accumulate quickly; a 0.3 single-scan P_d reaches roughly 0.97 after 10 looks. The range at which P_c first crosses 0.9 defines the cumulative detection range, typically 15 to 30% longer than the single-scan range, and it is the figure of merit search radars are specified against.

Category: Radar & Defense

Typical target P_c: 0.90

Combination rule: 1 − ∏(1 − P_d,i)

How Detection Odds Accumulate Over a Search

A surveillance radar rarely detects a target the first time the beam sweeps past it. At long range the returned signal is weak, the instantaneous SNR is low, and the single-scan probability of detection may sit at 0.1 or 0.2 against a fixed false-alarm threshold. What makes search radar effective is repetition: each frame revisits the same volume, and every look is a fresh, largely independent chance to cross the detection threshold. Cumulative detection probability formalizes that intuition. It answers the operational question, what is the probability the target has been declared by the time it reaches a given range, rather than the narrower question of whether any single look succeeds.

The mechanism that drives P_c upward during an engagement is the range dependence of received power. As an inbound target closes, received signal power scales as 1/R⁴, so SNR climbs steeply, and per-scan P_d rises from near zero toward unity over a relatively short stretch of range. The cumulative product therefore stays low while the target is far out and then sharpens rapidly as it enters the high-SNR region. This is why a plot of P_c versus range has a characteristic knee, and why specifying a radar at "P_c = 0.9 at R km" captures search performance far better than a single-scan number.

The fraction of scans that yield a detection at a particular range is the blip-scan ratio, and it is essentially the per-scan P_d measured empirically. Because the number of available looks depends on how long the target remains in coverage, the revisit time of the search frame is a first-order design lever: halving the frame time doubles the look count during the closing geometry and lifts the cumulative detection range without any change to transmitter power or antenna gain.

Cumulative Probability Equations

Cumulative detection probability (independent scans):
P_c = 1 − ∏_i=1^N(1 − P_d,i)

Equal-scan special case (constant P_d):
P_c = 1 − (1 − P_d)^N

Scans available during closing geometry:
N ≈ (R₀ − R) / (v_c × T_scan)

Per-scan SNR vs. range (monostatic):
SNR(R) ∝ 1 / R⁴ → P_d rises sharply as R decreases

Where N = number of independent looks, P_d,i = single-scan probability of detection on scan i, R₀ = range at search entry, v_c = closing velocity, T_scan = revisit time. Example: P_d = 0.3, N = 10 → P_c = 1 − 0.7¹⁰ ≈ 0.97.

Single-Scan vs. Cumulative Performance

Per-scan P_d	P_c after 3 scans	P_c after 5 scans	P_c after 10 scans	Scans to reach P_c ≥ 0.90
0.10	0.27	0.41	0.65	22
0.20	0.49	0.67	0.89	11
0.30	0.66	0.83	0.97	7
0.50	0.88	0.97	0.999	4
0.80	0.992	0.9997	>0.9999	2

Common Questions

Frequently Asked Questions

How does cumulative detection probability differ from single-scan probability of detection?

Single-scan P_d is the chance of declaring a target on one look at a given SNR and false-alarm threshold. Cumulative P_c is the chance of detecting on at least one of N successive scans: P_c = 1 − ∏(1 − P_d,i) for independent looks. As the target closes, per-scan SNR rises as 1/R⁴, so a 0.3 single-scan P_d reaches about 0.97 cumulative after 10 scans.

What is cumulative detection range and why is it longer than single-scan range?

It is the range at which P_c first reaches a specified value, usually 0.9, as the target approaches. It exceeds the single-scan 0.9 range because every extra scan adds an independent detection chance. With a 10 second frame against a Mach 1 inbound, the added looks during closing typically extend effective range 15 to 30% beyond the single-look figure.

How do scan-to-scan correlation and Swerling fluctuation affect the calculation?

The product rule assumes independent scan decisions. That holds for slow-fluctuating Swerling I and III targets that decorrelate between scans. For a steady Swerling 0 target with correlated RCS across scans, the looks are not independent and true P_c falls below the product-rule estimate, so designers apply a correlation factor or use measured blip-scan data to avoid overstating performance.

Related Terms

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Radar & Defense

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