Hitting Your Target

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A question that keeps coming up on the forums is why missiles and turrets have such different skill trees. With the Rubicon release, some of those questions will go away since the skills will undergo the teircide treatment, but the underlying reason will remain the same: because they're not the same. This is an entire accurate but completely meaningless answer that is usually tossed into the discussion when people feel that the poster could have just searched for one of the many threads on the topic. So what does it mean? How are they different?

This post will go through the main differentiator between missiles and turrets: their respective hit and damage calculation mechanics. A lot of this can be found in wikis around the web, but few (if any) of them go into the various components and what they mean in practical terms.

The obvious bits

Without going into the actual mechanics, there are some obvious differences that can be highlighted. The primary one is that turrets hit immediately on firing, whereas missiles need to catch up to their targets in order to hit. As such, turrets may or may not hit depending on what the statistical model says and how good your to-hit rolls come out, whereas missiles may or may not hit depending on whether the target manages to out-fly them or even destroy them in flight. As a side-note, drones offer an interesting mix of the two: they are damage dealers with a flight time that use the turret mechanics to determine whether they hit or not (yes, even sentries, although at 1m/s, they move a bit too slowly to catch up with anything and generally just rely on their long range to shoot things).

The Tracking Formula

Tracking Formula

The turret tracking formula generally refers to the “chance to hit” (or CTH) forumula, which is exactly what it says on the tin: the probability that the turret will hit its target. The formula consists of seven different parameters clustered into two main components and two subcomponents. Some of these parameters are characteristics of the ship that's being shot at and the others are attributes of the turret being used. The target ship parameters are:

  • Transversal — the target transversal speed.
  • Range — the distance to the target.
  • Signature radius — the target's current signature radius (effectively, the target's size as it appears to sensors).

The turret parameters are:

  • Tracking — the turret tracking speed, which tells us how fast the turret can rotate and still maintain a 50% hit chance.
  • Signature resolution — the turret's signature resolution (the optimal size of its targets).
  • Optimal — the turret's optimal range, meaning the range to target where the turret will start to receive range penalties.
  • Falloff — the turret's falloff range, which determines how quickly the chance to hit starts to drop if the target is beyond the optimal range.

The CTH formula combines and compares these parameters into two main components: the range component, which determines the chance to hit as a function of the distance to the target, and the tracking component, which determines the chance to hit as a function of the relative movement between the attacker and the target. The tracking component, in turn, consists of two sub-components: the movement and the size components, which (surprise!) alter the CTH as a function of relative movement and target size, respectively.

Before delving deeper, it's perhaps best to recall some basic algebra and note that the form xy+z can also be written as xy × xz, so it's actually trivial to separate these two components and consider each on its own. The chance to hit based on tracking is 0.5tracking component and the chance to hit based on range is 0.5range component — the total CTH is the product of the two.

Range component

The range component (max(0,range-optimal)/falloff)2 is fairly simple to understand: it determines how far into falloff the target is. If the distance to target is is less than optimal range of the turret, the whole thing comes out to 0; if not, it comes out as a percentage of that falloff range. What the tracking formula doesn't reveal, though, is that CTH is hard-capped at two times falloff range. Any beyond that distance automatically receives a CTH of 0%.

Still, we can already make some quick observations and illustrate them with simple numbers. Assuming that we have a turret with an optimal range of 1 and a falloff of 2:

  • If the target has a distance up to 1, the component comes out as 02 and the CTH comes out as 0.50=1 — in other words, based on range alone, you have a 100% probability of hitting the target.
  • If the target has a distance of 2 it is within half falloff (because (range 2 - optimal 1)/falloff 2 = 0.5). This means the component as a whole comes out as 0.52=0.25 and the CTH comes out as 0.50.25=0.841 — you have a 84% chance of hitting at this range.
  • At a distance of 3, you are at 1× falloff range and you therefore have a CTH of 0.51=0.5 — a 50% probability to hit based on range alone.
  • Finally, at a distance of 5, you are at 2× falloff and now only have a CTH of 6.25% (i.e. a component value of 22=4 and a CTH of 0.54). Beyond that, as mentioned, the CTH is hard-capped at 0%.

We can already observe that we want the component to be as low as possible — preferably 0, since that leads to the highest CTH — and that higher numbers in the numerator (due to high range or low optimal) are bad since they lead to a higher component value, whereas higher numbers in the denominator (falloff) are good since they reduce that value. Of course, these are all intuitive conclusions — of course high optimal and falloff are good and long distance bad — but this explains why this is case from a mechanical standpoint.

Tracking component

The tracking component (transversal/(range × tracking) × sigres/sigrad)2 is a bit more difficult than the range component since it is itself a combination of two sub-components: the movement and the size component. The key thing to understand here is that the size component is only really a (semi)fixed modifier to the movement: it makes that movement count for less or more than it otherwise would, but on its own, it is pretty meaningless.

Movement component

The movement component is written in a more elaborate form in this case because a lot of people keep talking about “transversal” and have it displayed on their overviews. The value that matters, however, is the angular velocity of the target, which is the same as transversal/range. This means that the movement component could actually be written as angular/tracking, which gives us a better idea of what we're really measuring here: the ratio of movement of the target around the attacker compared to the tracking value of the turrets.

Even so, it's important to remember that range is a factor in determining the tracking. Maintaining the same transversal speed at 1km and at 10km creates vastly different angular velocities so talking about transversal alone says very little. Also, whereas in the range component above, higher range led to lower CTH, in the case of tracking it leads to higher CTH. This helps us explain why “optimal range” is… well, optimal: because it's the range where we get no penalties from the range to the target, but get large benefits to our tracking. In practice, depending on how quickly the CTH drops off in falloff, the literally optimal range may be beyond this point because the range penalty reduces CTH slower than the tracking benefit increases it, but it's still a good rule of thumb. And anyway, from a mechanics standpoint, while range is a factor, the relevant comparison is between angular velocity and turret tracking.

If this ratio comes out as 0, we have perfect tracking. If the ratio comes out as 1, your tracking matches the target's movements and, barring size modifiers, you have a 50% chance to hit. This makes the angular-to-tracking ratio very much analogue to the range-to-falloff ratio described above, except that there is no hard cap on how high it can go. So a target that moves twice as fast as turret can track translates into a movement component of 2, a tracking component of 22=4, and a CTH of 0.54=0.0625, or 6.25% probability to hit, just like how a target at 2× falloff yields a 6.25% probability. But again, that's before we take the target size into account…

Size component

The size component is quite simply the ratio between the turret's natural target size and the actual size of the target. Generally, this is determined by the various size bands of the turrets: small turrets (meant to shoot frigates) have a signature resolution of 40m, which coincides with frigates having a signature radius in the region of 30–50m. Likewise, medium (cruiser-sized) turrets have a signature resolution of 125m and the intended cruiser targets have a signature radius in the region of 100–140m. Three guesses what the sigrad band of battleships is when we consider that large turrets have a sigres of 400m…

Again, this ratio between intended target size and actual target size only acts as a multiplier to the movement component described above. There are no zero-resolution turrets or infinite-size ships that could make this ratio 0, so the only way for the tracking component as a whole to come out as zero is if the angular velocity is zero.

What the ratio can do, however, is to compensate for fast targets. If we turn the tracking component on its head, we can look at the inverse ratio (sigrad/sigres) as a direct multiplier to the tracking value: a turret with a signature resolution of 125m tracks 3.2× faster than a turret with a signature resolution of 400m, assuming both have the same base tracking speed. Conversely, a medium turret (125m sigres) tracks 3.125× slower than a small turret (sigres 40m). So much like how transversal speed doesn't say much on its own, tracking speed is meaningless without taking signature resolution into account.

When we take into account what was said above about how quickly the angular:tracking ratio starts to yield near-impossible CTHs, we can quickly see how much of a difference these multipliers make. Simply doubling a 1:1 tracking ratio to 2:1 means going from a 50% CTH to just 6%, but having the wrong turret can multiply that ratio by 3 or more. Also, as with the range component, we can now get a good idea of what's good and bad for your CTH (and why): anything in the numerator — i.e. transversal (or angular) and signature resolution — is bad if it is high; anything in the denominator — range, tracking, and signature radius — is good if it's high.

Chance to Hit → Damage Modification

To reiterate, the final CTH is the product of the CTH from tracking and the CTH from range. Since the range to target is a factor in both of these, we have a situation where tracking is hugely important and range irrelevant at low ranges, and where tracking is nearly irrelevant and (optimal and falloff) range is hugely important at longer ranges.

Effectively, then, the two components act as dampers for each other. Even though the tracking CTH keeps increasing, it gets pushed down by the range penalties, and even though range CTH increases at the target comes closer, it too is pushed down by the tracking. Only with very long optimal in relation to the tracking value will we see the CTH level out before it starts dropping due to range.

The die roll and hit qualities

But all of this is just the calculation of the hit probability. The next question is how this probability is actually used. As can probably be expected in an MMORPG, this is done through a die roll. If the die comes out higher than the CTH, it's a miss; if the die comes out lower than or equal to the CTH, it's a hit of varying quality; and if the die lands somewhere in the first percentile, you score a wrecking hit.

The hit quality is a straight multiplier to the turret's damage, and is simply the die roll +0.5. Depending on how high that final multiplier is, it gets labelled with descriptions such as “scratch”, “solid”, ”critical” and so on. A wrecking hit, which you can only get by rolling ≤0.01, sets the damage multiplier to 3. There are a couple of effects to note here. As long as you have any chance at all of hitting the target — i.e. as long as it is within 2× falloff, you can always hit it with a wrecking shot for 3× damage. If you have a 100% chance to hit, your damage multiplier for a non-wrecking shot can only get as high as ×1.5; for 50% CTH, it maxes out at ×1.0.

All of this means is that low hit-chances makes your life miserable twice: not only do you have a low probability for a hit (obviously), but if you do hit, your multipliers can't get very high. Statistically, your average DPS for a given CTH can be determined by the formula: 0.03 + (CTH-0.01)×(CTH+1)/2.

A first thing to note is that, at 100% hit chance, your average DPS is actually 1.02× your base DPS: 0.99 average for die rolls between 0.01–1.0, plus 0.03 for the last percent of wrecking shots. Another thing to note is that damage drops off a lot faster than the CTH does, which in turn drops off a lot faster than any single range and tracking difference does. Except for at both extremes of CTH, where the fixed chance for a wrecking hit makes itself know, the average damage output percentage is always lower than the to-hit percentage. So even small changes in the input parameters can generate large changes in hit chances and even larger changes in the effective damage output.

  • At half falloff or half tracking, the CTH becomes 84%, and the effective DPS drops to 79%.
  • At full falloff or full tracking, the CTH becomes 50% and the effective DPS drops to 40%.
  • At full falloff and full tracking, the CTH becomes 25% and the effective DPS drops to 18%.
  • At double falloff or double tracking, the CTH becomes 6.25% and the effective DPS is down to 5.8%.

The Missile Formula

Missile Formula

The missile formula is at the same time both a much more straight-forward equation and a much more complex calculation. It is effectively a selection of the lowest possible damage output depending on the mix of speed and size compared to the missile's explosion radius and -velocity. Like the tracking formula, it consists of two main components — the stand-still threshold and the movement component — and of four sub-components. It still only has six parameters, however, and one of those could really be considered a constant under the current implementation. As with turrets, a distinction can be made between the missile's parameters and those of the target. The target parameters are:

  • Speed — the absolute speed of the target, irrespective of direction.
  • Signature radius — the target's current signature radius (effectively, the target's size as it appears to sensors).

The missile parameters are:

  • Explosion velocity — the maximum speed the target can travel before it starts to outrun the explosion.
  • Explosion radius — the minimum size the target needs to be to catch the full force of the explosion.
  • Damage Reduction Factor (or drf) — how much the combination of explosion velocity and explosion radius matters for the final damage.
  • Damage Reduction Sensitivity (or drs) — how sensitive the missile is to mismatches between its ideal target and its actual target.

Already here, a very important distinction between the missile formula and the turret formula can be made: the missile formula is not chance-based. It simply compares stats and immediately spits out a damage modifier that remains constant until the variables change. There is no die roll and no averaging needed to create a statistical answer for some probable outcome. Instead, it is simply a question of which is lower: the standstill threshold or the movement component.

Standstill Threshold

The standstill threshold — min(sigrad/exprad, 1) — gets its name because it sets up the best-case scenario of an immobile target and provides a threshold for how high the damage application can ever go. It is simply the ratio between the target's signature radius and the missile's explosion radius. If the target is smaller than what the missile is intended for, the damage becomes proportionally lower, and the ratio is capped at 1 so shooting a sufficiently large target means your missile does full damage and nothing more.

Movement Component

The movement component (expvel/speed × sigrad/exprad)^(log(drf)/log(drs)) is a bit more complex. It takes care of ships that move around faster than a missile can handle. It uses the same size component as the standstill threshold, combines it with a speed component, and then modifies those two ratios by a precision calculation.

Size Component

The size component looks and works much the same as it did in the previous instance, but with one important difference: it is no longer capped on the upper end. Instead, it is combined with the speed component so that a very large target can't simply compensate for that size by also being very fast. Had the ratio been capped, a battleship travelling at 1km/s would have received the same damage reduction as a frigate moving at 1km/s. Instead, the 10× larger battleship also has to travel 10× faster to see the same reduction.

Speed Component

The speed component is simply expvel/speed — the faster the target is compared to the missile's explosion velocity, the less damage it takes. Again, since neither of these ratios are capped, being fast isn't enough and can easily be cancelled out by being too big. But just because there are no caps in this part of equation doesn't mean that damage will suddenly go into the stratosphere. Remember that the formula as a whole always picks the lowest possible number, and that the standstill threshold already sets 1 as the highest value it will ever spit out. So even if the speed and size components would combine into a very large number, the final multiplier would still only be 1 since that will now be the lower value.

Precision

The precision component of the missile formula is where we get into the properly obscure and esoteric parts of the game. Its value of log(damage reduction factor)/log(damage reduction sensitivity) acts as an exponent for the combined speed and size ratios, and effectively establishes a damage falloff curve for when the target starts to outfly the missile.

In the current implementation, the DRS is effectively a constant: all missiles have it set to a value of 5.5. I say “effectively” since it is still an attribute of the missiles and not a fixed part of the formula, so it could potentially be altered for individual missiles. However, since the same effect can be had by just changing the DRF, there's no real reason to do so.

Neither of these attributes are something you will see on a daily basis and their impact is not immediately obvious. You generally have to dig into unlabelled entries in the item database to find them (see for example the Scourge Fury HML data listing). The short and sweet of it is that a higher DRF means that the missile reacts worse to being outmatched by the target. If the DRF is the same as the DRS, we have a perfectly proportional missile: if the target flies twice as fast as the explosion velocity, the damage is halved; if it flies four times as fast, the damage is down to one quarter base damage etc. If the DRF is lower than the DRS — and it pretty much universally is for the current set of missiles — then the damage reduction is less drastic as targets get smaller and faster. If it were higher in any missile, the damage reduction would grow increasingly severe the faster and smaller the target was.

From a purely mathematical standpoint, this should come as no surprise: the movement component only matters if it is lower than 1, otherwise the standstill threshold is the one that counts for the final result. So the base number is higher than the DRS, the exponent is >1 and the final result comes out lower than than the base value, so the speed and signature has even more effect than is immediately obvious.

For a given missile type, the DRF is generally used to create a different damage application characteristic for the T2 variations of that missile. For instance, the high-damage “Fury” missiles have a higher DRF than their T1 counterparts, and the “Precision“-type variants have a lower DRF still. When combined with the differences in explosion radius and velocity that those variants have, they often take the supposed advantage or disadvantage of the missile and pushes it even further than the base stats would suggest. So “Fury” missiles don't only have a higher explosion radius and lower explosion velocity — making them a poor choice against a smaller and/or faster target — they also have a higher DRF that makes smaller and faster targets take even less damage than they otherwise would.

Practical Consequences

The practical consequences of all of this boils down to one key difference between missiles and turrets: missiles are almost completely devoid of pilot interaction compared to turrets. Aside from the speed of the target, almost everything else is decided by the time the two ships undock from their stations. Ok, that's maybe a bit of an exaggeration — there's still the application of target painters, propulsion mods, and their counters that may alter the signature radius as well, but the actual flying makes no difference. At best (worst?) the target may manage to fly too fast, too far for the missile to hit but if it does, only one (and a half) of the parameters involved are properly variable.

This has both its benefits and its downsides. The main one for both, depending on your perspective, is that the only way to compensate for your target's attributes when you pick a missile ship is to ensure that you fit it properly. Be it through rigs that rectify some horrible missile stat or through ewar that brings the target into the missile's preferred envelope, the missile has to fly itself onto the target and the piloting of the missile ship itself makes no difference.

Turrets, by comparison, are completely interactive every step of the way. The ability to hit and miss is not even remotely as binary as with missiles, but rather sit on a continuum from “not a chance” to “guaranteed”. Even the damage itself is not as clear-cut: even a perfect shooting opportunity can yield a horrible hit and a horrible firing solution can yield a good a not entirely awful hit. Moreover, the way you pilot your ship in relation to the target can make even the most completely outmatched turret hit at full accuracy (or, from the opposing point of view, even the most spectacularly tracking turret can be outflown). A tracking enhancer forgotten in the hangar can be compensated for with good flying, but a rigor rig cannot.

So even though both weapon systems share many of the same variables in terms of what matters in combat — speed, range, size — and even though some attributes may look like each others' counterparts — explo.velocity for signature resolution or DRF for falloff, for instance — the way they interact with each other and the results they yield are vastly different. The good news is that, the next time you see the forum platitude of “missiles are not turrets”, you will know what on earth this actually means…