Probability of a ballistic missile salvo hitting US Navy surface ships

On a recent Midrats podcast episode, Gerry Doyle and Blake Herzinger discuss the likelihood of a Chinese ballistic missile like the DF-21 successfully hitting a surface ship. Hitting a moving target at sea makes for a hard targeting problem, as perhaps evidenced by the lack of Chinese propaganda claiming successful at-sea tests, except perhaps stationary barges. It seems the Chinese have moved back to land-based testing with targets moving along a track. Here’s an interesting bit on the probability of kill:

Nobody publicly will say exactly what size plot of the ocean a [ballistic] missile seeker can look at, but you can crunch some numbers. If the ratio of the missile’s kill radius, how much it can see and hit with that seeker, to the targeting location, which is a much larger area where the carrier can be — let’s say the ratio is 0.5, a missile can see and locate half of where it can be, to get to a probability of kill of 0.6 or 0.8, you’re talking dozens of missiles. And that is before taking into account countermeasures .

That’s not exactly right. If the radius of the seeker is half that of the potential target location, then it is looking at about 25-30% of the surface area where the ship could ultimately be. That’s because the area of the circle grows along an exponential curve with the radius. Below is a figure from the RAND US-China Military Scorecard report which was being referenced.

For example, if a carrier can move at 30 knots, and the time from missile launch to the target area is 20 minutes, then you have roughly a 10 mile radius the carrier could be. They point out that there could be a much longer delay in real life between target acquisition and launch. But a 10 mile radius translates into a search area of 314 square miles compared to the seeker search area of 78 square miles if it has half the radius (5 miles).

So to reach an 80 percent probability of hitting a surface ship, with a missile kill radius-to-targeting location ratio of 0.50, you’d need to launch 13 missiles. But that assumes a 100% kill probability once a missile acquires the target. The final piece to the puzzle of course is once the seeker acquires the target, what is the percent likelihood of a kill? If landing the shot is only a 40% likelihood, you’d now need 55 missiles.

Below is a nice table, again from the RAND US-China Military Scorecard report. Some definitions are below.

• Missile kill radius, or the maximum distance from its prelaunch aim point within which the missile can still engage (find, locate, and maneuver to) the target (see Figure 7.2). The kill radius (or basket) depends not only on the missile’s kinematics but also on its in-flight search, identification, tracking, and data-link capabilities. Published reports speculate that the DF-21D kill radius might be between 25 and 40 km.5
• The targeting location CEP is the expected difference between the actual target location and the targeted location at firing time.54 As noted previously, OTH radar has a targeting location CEP of between 22 and 178 km, depending on the distance from the target and other variables, though delays in transmitting identification and location data to the firing elements may effectively add to the CEP.
• The weapon Pk is the probability that the weapon will hit the target given that the target is within the kill radius of the missile’s aim point when the weapon arrives. It is a function of the weapon’s capabilities (e.g., speed, reliability, maneuverability, sensors) and the target’s defenses, both kinetic and nonkinetic.