Why They Are Nervous. Short Excurse Into the Modern War. Part 4.

Now to the most important part. Recall what I wrote about ASW and missile attack in the previous three posts? Now forget it all because today it matters only under some extremely limited circumstances. Here is a simple illustration about CBG's relative scale to modern ranges. 

In this case, as you can easily see, it really doesn't matter how you configure your ASW for CBG, we don't even talk about a group of ships on their own, with only MH-60R Seahawks as air ASW assets at their disposal. Those are magnificent and advanced machines but their problem for modern ocean-wide warfare is the fact that they have a speed of 270 kilometers per hour and a maximum range of 830 kilometers, that is the range at which they can fly in ASW configuration in a straight line, but as you can already surmise, ASW is a very energy costly affair (with all this hovering and dipping at the Datum), granted that this Datum is (was) provided by good ol' Soviet Charlie-IIs and Malakhits. 

This time is over. Introduction of P-700 Granit with its range of 600 kilometers and fully networked salvo of these missiles in early 1980s was the thing which finally transferred ocean warfare from a combination of ASW and Air Defense to almost strictly Air Defense problem, granted huge Project 949A (Oscar-IIs) SSGNs could be tracked by the US SSNs at the ranges beyond CBG's ASW ranges. Sometimes they could track Oscars, sometimes they couldn't. But there was one thingy about those pesky P-700 Granits--unlike with Malakhits and other older anti-shipping missiles, there were a significant chances to miss a Flaming Datum when Oscars launched their Grantis. Once they were airborne it was entirely up to E-2 Hawkeyes to detect such a salvo and vector on-duty F-18s and F-14s for an intercept. Let's fast forward a bit, P-800 Onyx emerges and while officially its range is 660 kilometers, it is a much more advanced missile and its latest versions are reported to have a range in excess of 800 kilometers and speeds in excess of Mach=3. Look above at the illustration. What's left then? How to deal with this threat? Right--one needs Aegis and Standard missiles which theoretically can handle a salvo of...P-700 Granits. Theoretically. 

But P-700s are old missiles, what are you going to do against Onyx with submerged start? Welcome to a missile exchange paradigm and welcome to Salvo Equations. But since we discuss a real war not some fanboys' rah-rah bullshit from Popular Mechanics or Forbes, we have to recall that the war is a probabilistic event. So, what are, then, ways both warring sides approach and tackle this problem? Remarkably they are very similar because the same math and physics is used. For a commanding officer of ANY sub in the world carrying anti-shipping (or even TLAM) missiles the first thing which comes to mind is the assessment of Probability of a Success of whatever his sub is going to do, in our particular case--shooting and destroying a surface target(s).  In old times poor lads and their navigator and weps guys would calculate this probability by hand, by calculators etc. Until Combat Informational Control Systems or Battle Management Systems (Computers) which started to appear in their advanced forms precisely starting from the times mentioned many times before Charlies started to deploy, they started to calculate those probabilities based on the fusion of incoming data ranging from targeting to tactical and operational data from many sources (sonar, comms, visual, radar etc.). 

As we all know from the basic probability course, the probability of just about any event is calculated by multiplying probabilities of many related events which must occur in order for us to have a shot at achieving what we set to achieve.    

As you can see yourself a lot has to be done to kill a target. Even if you have a probability of detecting a target Pdetection =0.95, and the reliability of our targeting Ptargeting=0.95 and we have a probability of remaining undetected by enemy equaling 0.9 and whatever other probabilities factor in our probability of killing our target Pn=0.9, we will get: Pkill= 0.95 x 0.95 x 0.9 x 0.9 = 0.731. These are not the best chances but they are solid. We would love our chances to be within 0.9-0.99 (wink, wink) range. So, how do we increase our chances? Range and speed and excellent combat training. 

 

Once tactical characteristics of 3M22 Zircon have been confirmed, a man, who unlike me has an impeccable background in submarine warfare, Captain 1st Rank Igor Kurdin, who is usually very reserved about weapons, went on record: 

«Дальность «Циркона» составляет от 500 до 1000 километров. С такого расстояния никто нашу подлодку не засечет, если только за ней не следят давно и целенаправленно», – рассказал председатель Санкт-Петербургского клуба моряков-подводников ВМФ, капитан 1-го ранга Игорь Курдин. Собеседник напомнил, что задача американских противолодочных сил – засечь нашу АПЛ до того, как она произведет ракетные пуски. «Гиперзвуковая скорость предполагает, что после пуска «Циркон» практически невозможно уничтожить», – добавил он. 

Translation: “The range of the Zircon is from 500 to 1000 kilometers. From such a distance, no one will spot our submarine, unless they have been tracking it for a long time and purposefully, "said the chairman of the St. Petersburg Navy Submariners' Club, Captain 1st Rank Igor Kurdin. The interlocutor recalled that the task of the American anti-submarine forces is to detect our nuclear submarine before it launches missile launches. “The hypersonic speed suggests that once the Zircon is launched it is almost impossible to destroy,” he added.

We look again at the illustration in the beginning of this post and take in the range of 800 km. If such a range is viewed for regular Onyx with its varying speed on the flight path in excess of Mach=1.5-2.0 with acceleration to M=3.0 at the terminal, we are looking at the average velocity of M=2.0, which gives us Time = 800 km/2450 km/hr = 0.32 of an hour or roughly 19-20 minutes. With Zircon and its average of Mach=9 we are looking at 800/11025= 0.072 of an hour, which is around 4 minutes. No system can react to this, nor there are any air defense means which can intercept such a missile in any quarters, be it head on or, let alone, in pursuit, granted it is even detected, not to speak of a Datum which may stop being a Flaming one. This is for only one 3M22, when they fly in salvo of 2 or more...take a guess.  It comes down to merely reliability of missile itself, its ability to perform strictly on its technical merit which defines the probability of a kill. Here is what I wrote three years ago in my book:

A going and reasonable assumption today in regard to the latest anti-shipping hyper-sonic weapons is that the probability of intercept of such weapons, capable of Mach=9+, well in excess of any existing anti-missile weapons, even without maneuvering on terminal approach, is statistically insignificant. That is, in a basic Salvo Model for the losses of attacked force:

Where b1 is staying power of ships in enemy force B, which is the number of the missiles required to take out of action those respective ships. It takes a single missile of such a class to put any large combatant, with the possible exception of a nuclear aircraft carrier, completely out of action, thus making b1 =1. Coefficient b3 denotes a defensive power of the ships in B, which is the number of good enemy shots which will be destroyed or deflected by the defender—there is really no objective evidence of modern AD systems being capable of intercepting hyper sonic missiles. This makes b3 = 0 and, consequently, makes the multiple of b3 B = 0 ; this is a definition of a turkey shoot, in which the attrition ΔB of the opposing force, depends strictly on a number of hyper sonic missiles in A’s salvo at B. In other words, the equation is reduced to:

because  b1 =1 Alpha (α ) denotes the striking power of A, which is the number of missiles which would hit the target if there were no defense. There is no defense currently and this effectively eliminates B as a player in case of a missile exchange with A, thus making the exchange mostly a matter of reliability of missiles themselves. 

We will review those salvo transformations later, but for now a juicy piece of math with formula for a salvo by a group of subs, or missile carriers on the surface, just tweak some coefficients.

Pay attention to both Qs. You can easily find the textbook The Submarine Tactics by Captain 1st Rank Khvosh in Russian and, of course, intellectual tour-de-force by legendary late Captain Wayne Hughes and his Fleet Tactics and Coastal Combat. 

 

To be continued...

Why They Are Nervous. Short Excurse Into the Modern War. Part 2.

Now, since there was a consensus on me continuing with this ASW thingy, immediately--THE clarification. On the illustration with circles-ranges of Russian subs along the Atlantic coast of the United States (in the previous post) we view the situation from the point of view of the known position (and even Flaming Datum, which IS discovering sub's position by it blowing its cover) and immediately can draw circle-ranges and, in reality, this is how those aggressive humanity-hating, vodka-drinking Russkies will see the map. Not so in Pentagon. Before ANY salvo and, consequently, Flaming Datum (and even this must be discussed with some huge caveats), what American ASW people will see before the hostilities start will be this ugly picture. 

No circles here but only green shaded areas limited by a range of 500 nautical miles. I believe CBO from where the template is taken meant 500 regular miles, but let's assume that these are nautical miles which gives us 1.852 x 500 = 926 kilometers, precisely the range at which 3M22 could be launched in case of (God forbids) war. So, you may continue to mentally fill all those ranges with green thus reaching, eventually 2,000 miles range and beyond where latest mods of 3M14M with the range of 4,500 kilometers could be launched. 

So, looking at this green mess one has to asks a question of how many and where those Zircon-carriers are. If there is no war, but it already begins to smell funny in the air, the only thing you can do is to "pack" these green areas with all available ASW assets without losing readiness for war. Well, while ships in ASW role are fine and dandy, they really are not that great if it comes down to a colossal area such as shown on both coasts. Modern submarine will hear those guys way earlier than they will be able to detect them, and, considering today astonishing advancements in quieting and an "open" water theater both at the Pacific and Atlantic coasts, the sub will simply avoid detection by maneuvering. So, call Jacksonville (VP-8) and also request additional Virginias to run the "perimeter" in hope to run into those pesky Yasens or Oscar-IIMs to prevent them from launching. Well, the big game starts. 

The US Navy is a proud and, despite its problems, still very powerful force. And although a number of credible ASW scholars warned for decades that ASW of the US Navy is a "soft spot", even the ensign fresh from USNA knows that you need to guard US Navy's most important installations and bases. Like Norfolk. So, what do those people do? They take the compass and start choosing the places with the highest Probability Density for the hostile sub's position. I omit for now this rather not difficult issue because at this stage of the warfare we are talking about not tens or even hundreds of kilometers of ranges, no, we are talking about thousands.  This is unprecedented in history. Mind you, Russia has already Zircon with the range of 1,500 kilometers and speeds in excess of Mach=12-13 in works. So. You get this (simplified)--you draw the baseline and then draw (with Norfolk in center) a semi-circle which matters a great deal, especially its shaded ocean part, which stretches (has radius) 1,000 kilometers into the Atlantic. 

This is where your aggressive, democracy-hating, Russian Yasen or Oscar-II M SSGNs will be, probably. Yes, a good term to use in this particular case--probably. How much is this "probably"--later. Now everything gets easy, not. The area of this probable Russkies' "station" is just one half of this circle: A=0.5 x 3.14 x 1000^2= some measly 1.57 million square kilometers. Easy-Peasy (I kid, I kid). Now, once ASW/Patrol aviation enters the picture--mind you, we are still not shooting at each other--we can start calculations in earnest. 

1. What is an Operational Sweep (rate) for a single, say, P-8 Poseidon, or rather a drone, using its main initial detection instrument of MAD--Magnetic Anomaly Detector. As we assumed previously, we give the width of Poseidon's MAD of 2 kilometers, we also assume for Poseidon a slightly higher than venerable P-3 Orion's search speed (P-3 flies at 360 kilometers per hour while "sweeping"), just for the sake of experiment, let it be 400 kilometers per hour. That makes Poseidon's sweep rate, while it will be conducting a Random Search, or SR = 2 kilometers x 400 kilometers per hour = 800 square kilometers per hour. Now we are on our way. 

2. For a single P-8 Poseidon to sweep these "measly" 1.57 million square kilometers area thus requires: 1.57 million sq.km : 800 = only 1,962 hours or about 82 days. I told you--easy, right? Of course, common sense tells us that more than one Poseidon will get involved, but as you already saw previously, unless we are talking about a Flaming Datum (that is either actual detection or sub giving its position up) we are facing Probability of Detection of such sub for a theoretical single P-8 Poseidon as:

In this simple formula our C is just the ratio of our effective search rate or sweep rate (800 sq.kilometers per hour) to the Area to be searched. Which in itself is not a probability of detecting a sub in the area but only a probability of our sensor (MAD) to detect it if it is there. 

We start with an easy example: let's say our Poseidon needs to search the area of 800 sq.kilometers. Our C thus is 800:800 =1, so:

Second sweep over the same area then gives us: 

                POD=1-0.37*0.37= 1- 0.1369 = 0.8631 (Respectable)

The problem, of course, is that once the Area of search increases two fold, to 1,600 sq. km (C=0.5), our POD drops in a single sweep to 0.39. And once we get Poseidon on station sweeping the Area of 4,000 sq.km (C=800/4000=0.2) we will get only POD of 0.18 after 5 hours of sweeping. So, imagine now that in our case we are talking about the area which is THREE orders of magnitude larger. Of course, naturally, it will be assumed that those Russian subs will be abusing the relative safety of farthest edges of this circle. Possible? Possible, but then again--they may simply sneak in and out, or just close in for a launch at 500 kilometer range. What one has to remember, that even from the range of 950 kilometers it takes a salvo of Zircons flying at M=8 to reach the target only 5.5 minutes. And this simple fact throws all modern ASW tactics into a full blown chaos, because any carrier of 3M22 can afford to make a gift of a flaming datum to an enemy, without much risk for itself. Why it is so--later. 

Update: thanks to readers--US used P-8 Poseidon, unlike those of India's Navy, doesn't have built-in MAD and "theoretically" is supposed to operate MAD carrying drone. Well, good luck with that. In the same time, good ol' P-3 Orion still has some life left. You can recalculate for P-3 with its sweep speed, at 60 meters altitude, of 360 kilometers per hour.   

To Be Continued...     

Why Mathematical Models Break Down. And Why Our Life Depends On That-5. What Is Probable.

So, consider now a simple problem (such as it is taught in War Colleges in Russia) for a simple tank (or even ship) scenario of shooting at target. This is an actual problem and it is excellent for demonstrating how non-linear war is. Recall, we already know a little bit about Osipov-Lanchester equations, whose solution is quadratic. So here it is:

In a combat the blue tank (ours) detects red tank (bogey) and the blue crew is given an order to dispose of bogey red tank in three shots max. You may apply this to, for now, ship on ship artillery duel.  Here immediately comes a scientific assumption—we, for the simplicity of experiment, assume that we already know the actual probabilities of each of the three shots. Those probabilities will be described by a much complex interaction (will see how in Salvo Model) of crew's level of training, ballistic computer properties and myriad other things, which are not the point now. So, say we know that the Probability P₁ of the first shot hitting the target (red bogey) is 0.6, consequently P₂=0.75 and P₃=0.85.  We also know our crucial Omega, ω, a mathematical expectation, or, speaking in layman's lingo—the weighted average of hits required for disabling such type of a target as our red bogey. Say, our ω=1.2. So, for this particular tactical task what will be the probability of killing the (red) bastard? It is warranted to say that Probability of this "kill" is also THIS very important and dominating parameter which defines what is known as a decisive element of any commanding decision, be it on one-on-one battle or in a very complex, multi-level operation—Criterion of Effectiveness. Criterion of Effectiveness is, most of the time, a probability of killing enemy SOB thus completing the task and attaining our objective(s).

The solution is very simple:

For this kind of task the Probability of a "kill" P_k from three shots will be:

 We plug in our numbers and get:

  

So, the probability of out tank killing the evil bastard is very high and pleasant 0.95. Good job! Death to occupants! 

Of course, one can also go exactly 180 degrees and using desired probability, say I want to defeat SOBs who deployed their MLRS launcher about to blow us up with the probability of P_d = 0.97, which is THE Criterion of Effectiveness, and get the number of required forces (tanks, ships, missiles etc.), aka in Russia as Naryad Sil (literally—forces which are "dressed"). Of course, in this case one will have to consider such things as probabilities of, say, tanks hitting the enemy with the first shot. So, say in this case, we need to count how many tanks we need to blow enemy's missile launcher (MLRS) up and then repulse enemy's counterattack. If we consider that the probability of hitting the target with the first shot will be the same (for simplicity of demonstration) for all tanks and is P_s = 0.45, we, using good ol' formula (in real life it will be much more expanded and complex one, we'll get to that too):         

Lower case n here is this number of tanks which we have to solve for.  Solution is easy:                  

After using logarithms (not crucial for now) we get our n=5.87 or, rounding it up, 6 tanks to do the mission. 

These are simple examples. But, as I stated, these examples are easily applied to other forces and here is the trick. These things, and much more, used to be done by staffs—they still are doing this and models and mathematics they operate is very complex. Nowadays these are computer battlefield networks which do most of this job but here is the deal—commanding officer, operational staff continue to matter immensely and especially on a purely human level. Greatest military minds seldom calculated themselves what is above and much more, but secret to their success was having those highly developed synapses which allowed them, very often without calculator or logarithmic ruler, see a larger tactical and operational framework. They had this tactical-operational non-linear intuition which helped them time after time achieve success in a seemingly completely chaotic business of war. From that, also, all Combat Manuals, Tactical and Operational procedures were written. It is true when they say that military manuals are written in blood. War is probabilistic in nature—always was—and that is what many laymen fail to recognize, that excellent tactical and operational level officers are developed in a tremendously rigorous mathematical (and physics) fundamental sciences, which allow them to proceed further into an extremely complex world of modern military technology and its combat use. Without understanding of modern high tech combat and being able to predict outcomes within highly non-linear combat framework no serious discussion is possible...

To Be Continued...