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(Somewhere in the Northern Pacific)

Sonar to Conn:
“Hold new sonar contact, designated Sierra Two, bearing 073, no aural classification. ”

OOD to Captain:
“Sonar reports new contact, designated Sierra Two. ”

Sonar to Conn:
“Sierra Two classified as possible close-aboard submarine. I’ve got steam noise.”

OOD to Captain:
“Have commenced active sonar search. Currently hold no active sonar contacts. Sonar is classifying Sierra Two as possible submarine. Am assigning passive tracker, bringing the ship left — Helm; left degrees rudder, steady course 340.”

Captain to OOD:
“Take that rudder off, you’ll give our position away.”

Sonar: “I hold planes’ transients, bearing 250. Possible submarine.”

OOD: “Ready weapon in tube two. Captain, I’m making weapon ready, tube two, probable submarine off my port hand.”

Captain: “Captain has the conn. Status of making weapon ready?”

Fire Control: “Weapon ready, tube two.”

Captain: “Fire tube two.”

Sonar: “Conn; we’ve got launch transients, bearing 065. Torpedo in the water, bearing 065.”

Fire Control: “Captain; Weapon ready, tube one”.

Captain: “All ahead flank. Helm; Left full rudder, steady course 300. What’s his bearing now?”

Sonar: We’ve lost track Sierra Two. All we hold is the incoming weapon.

“It is possible that no environment so severely imposes the ‘fog of war’ as the undersea battleground for SSN versus SSN engagements. Information, real time information, and the ability to exploit the data will ultimately determine the outcome of any engagement.

The particular problem of torpdeo evasion is a complicated one, made all the more so by the extraordinary pressures of submerged combat. What to do with the submarine, how to employ the counterfire weapon, when and should the unit under attack attempt to re-engage? These are all questions the submarine’s Commanding Officer must answer within seconds of commencing battle.

This problem can be examined through the use of an idealized engagement scenario. An imaginary aggressor has at his disposal a very simple weapon — a torpedo of unlimited endurance. It acquires and homes on its target by active acoustic transmissions. It relies on its acoustic returns to effect search, localization, and terminal homing. The torpedo is also gifted with the ability to home irrespective of target evasive maneuvers. It has only one flaw. The acoustic environment being what it is, the possibility exists that before acquiring the target the weapon may begin homing on a false target -possibly a decoy. Further, the weapon can terminally home on only one target. That is, once homing begins, whether on the target or on a false return, the weapon is locked in and unable to disengage.

At this point a bit of underwater acoustics is in order. If an active emitter is operating at a frequency of, say 100 KHz, then there will exist significant reverberations at and around that frequency. These reverberation sources include the ocean bottom, the surface, waves, biologics, eddies and currents, and many others. The performance of a processor whose task is to recognize the returns from a contact which are valid, will be degraded by the reverberations. If, however, the returns are shifted in frequency away from the emitter frequency, discriminating valid returns from false contacts becomes an easier task.

The typical mechanism for evaluating return frequency shifts is the well known doppler effect. That is, if the contact of interest is closing or opening in range, the active returns will be shifted up or down in frequency in direct proportion to the range rate of closure or opening. This “doppler shift of convenience” proves to be a great aid in avoiding false contact returns.

The torpedo described above has given the user of the weapon a simple tool, i.e., it may be designed to filter out that frequency region most affected by active reverberations. In doing so, it has in effect decreased its probability of false contact acquisition. There is of course a downside tradeoff. In filtering out the frequency region in and around the emitter’s center frequency it has made itself blind to zero doppler (little or no opening or closing range rate) targets. Thus, the Commanding Officer who uses this hypothetical weapon opts to have it either filter out near-zero doppler returns or process the entire spectrum of incoming data.

The evader in this simple problem has a choice of two evasion gambits. Once alerted to the incoming weapon he may either evade away (good doppler) or across the line of sight (no doppler) of the incoming weapon.

Having described this rather idealized scenario, we are faced with developing an optimal strategy for both the aggressor and the evader.

Which strategy would you employ if you were the aggressor? The evader?

The simple game described above may be generalized as follows. Two players, operating to achieve diametrically opposed goals, play under a very rigid set of conditions. While the rules and possible strategies are known to both players, the option actually chosen by either player is unknown to the other.

Option 1 for player I, the aggressor, is to employ active doppler filters (doppler enhancement) in his torpedo; Option 2 is not to employ these filters (no doppler enhancement). Option 1 for player II (evader) is to evade away; option 2 is to evade across the line of sight of the torpedo.

What we would like to do is decide on an optimal strategy for both players – recalling that the hypothetical torpedo has infinite range. The mathematics for this engagement of a homing torpedo versus an evading target shows:

View full article for table data

This shows that with doppler enhancement used, the evader running away is always hit, but without doppler enhancement the evader running away will be hit only 90% of the time. On the other hand, when doppler is depended on a crossing evader has zero doppler and no chance of being hit. Player IT, the evader, will never choose evasion option 1 – to evade away. Doing so guarantees that at a minimum the probability of being hit is 0.9. If, on the other hand,, option 2 is selected, you (evader) are certain that at worst the probability of being hit is 0.7. It may even be zero if player I chooses doppler fliters.

Now consider the game from player I’s point of view. He is trying to maximize the hit probability of his torpedo. For the game depicted it is clear that he will always choose option 2 (no doppler enhancement).

In summary, the game reduces to the following: Player I, the aggressor, should always choose no doppler enhancement, while player IT, the evader, should always choose to evade across the line of sight. This particular class of zero sum game is called stable. The value of the game is said to be 0.7.

Generally, the solution to such games is a probability distribution. That is, player I should choose option 1, X% of the time and option 2, Y% of the time. Likewise for player IT. In this special case, player I chooses option 1 0.0% of the time and option 2 100% of the time. Likewise, player 2 chooses option 1, 0.0% of the time and option 2, 100% of the time.

Now lets assume that the probability of false contacts increases with the distance run by the torpedo when doppler enhancement is not selected. Thus, if player n chooses to evade away, the total active torpedo run will be greater than the total torpedo run for evasion across the line of sight.

Moreover, for the simple case proposed, the probability of a torpedo hit, given no doppler enhancement, is greater for torpedo evasion across the line of sight than it is for an evasion in the line of sight. We have arrived at a point where we can solve a more realistic formulation of the doppler enhanced torpedo evasion problem. The aggressor has opted to engage the evader at a range of five Kyds. The weapon has an operating speed of 50 knots. The evader has a speed of 30 knots. We assume a value of 0.10%/Kyd for the probability of false contacts. Having determined run of the torpedo, if player I chooses no doppler enhancement and player II chooses to evade away then:

View full article for table data

What does this solution tell us about the very simple evasion problem described? First, it is clear from the results that there is no simgle correct strategy for either the aggressor or the evader. In fact, the aggressor should shoot with doppler enhancement selected 24% of the time and without doppler enhancement 76% of the time. The evader should evade away 46% of the time and across 54% of the time. In the final analysis the aggressor has a 46% chance of successfully engaging the evader.

Or we might consider the results from the viewpoint of the enemy force commander. He has at his disposal 100 SSNs. At some particular moment all come under simultaneous attack. If his unit commanders all choose to evade away and the aggressors all employ doppler enhanced weapons, then the force commander expects to lose all his assets. This is the scenario where the aggressor is aware of enemy force doctrine and attacks to exploit that knowledge. If however, the enemy unit commanders are instructed to draw a card at random from a deck numbered one to 100, and act according to the evasion rules found above, then the enemy force commander will expect to lose only 46 units. While there is an element of the ridiculous in carrying out the optimal solution, it is clear that for the hypothesized scenario a mixed strategy will clearly result in fewer loses than a single strategy.

Questions are raised by these results: first, of course, how valid is the model employed? Clearly many factors have been left out. For instance, current weapons do not have an unlimited endurance. Nor are they able to be accurately steered onto the target in all instances. The evader modeled is also deficient — his ability to evade outside the torpedo’s acoustic cone has been discounted. How does the employment of torpedo countermeasures affect the false contact model proposed? There are many other shortcomings.

However, the reader should also remember that the scenario was kept simple, but a careful look at the problem reveals that, for the situation modeled, the ordering of the payoff matrix is probably correct. That is, while the numbers used might not be exact, the order in which they appear makes some logical sense. If the ordering is correct then the reader can convince himself that the optimal strategy will always be a mixed strategy. That is, there is no one attack or evasion strategy which is best. Best is always some combination of the two!

There are many interesting tactical questions related to this model. At what depth should torpedo evasion take place? (Clearly it depends on where the aggressor has placed his weapon). If the aggressor knows, for instance, that tactical dogma for the enemy is to evade deep, then shouldn’t all torpedoes be placed below the thermal layer? And if all weapons are placed below the thermal layer, shouldn’t there be a certain percentage of the time when the evader should go shallow?

Tactical decision making is best done by man, not by machine. The subtleties of each particular engagement do not lend themselves to mathematical modeling or optimization. For instance. do we factor in the Commanding Officer’s knowledge of the particular enemy commander’s reaction to an attack?

Just as dangerous, however, is unthinking adherence to tactical dogma. An exercise like the one just completed above, forces submariners to examine the tradeoffs affected by the various options available. The cost of adhering to “the right solution” is lost ships and lost Jives.

While certainly no panacea, the examination of various tactical desisions under the game rules described above will clearly enhance the commander’s understanding of the situation.

P. Kevin Peppe

Naval Submarine League

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