Marco Santarini’s “Bismarck and Hood – The Battle of the Denmark Strait – A Technical Analysis For A New Perspective” delivers what it promises, and does indeed provide a new perspective on this well-known – and often poorly interpreted – engagement, providing what is, at least to this author’s knowledge, the first thorough and professional statistical analysis of the battle.

Attempting to analyze a brief action – in practical terms almost a skirmish – via statistical methods requires both skill and courage. The very essence of statistical treatment is that it works best when the sample base is large, and when relatively unusual events are few. The statistician can, fore example, predict with some confidence how many heads or tails might come up after a thousand tosses of the coin, but is helpless to predict the results of the very next toss. Although the statistician can predict unusual outcomes – e.g. the presence of ten ‘heads’ in a row -- but cannot begin to define exactly where in a sample such a specific sequence might might take place. The initial question then becomes whether or not a purely statistical approach to this action is appropriate at all. It is.

Using mathematical means to simulate and analyze combat has been around for a very long time. Most war games are in fact based upon mathematical models, and in the final analysis most mathematical modelers are, in some very real sense, wargaming. The construction of a typical mathematical model has become, by now, a fairly straightforward process. The most difficult aspect of the problem was, and tends to remain, the development of a system that adequately addresses the attrition due to target damage.

There are, basically, two approaches to this, which Carlson defines as ‘derministic’ and ‘stochastic’. The deterministic model (which Santarini calls ‘erosive’, a term I like better) is most clearly typified in the U.S.N. Naval War College’s Maneuver and Fire Rules and in the well-known Fletcher Pratt Naval War Game. The erosive model essentiallkly sees damage as cumulative, basically ‘the death of a thousand cuts’, none of which is individually critical, but each of which slowly depletes the target’s fighting capability. The stochastic model, in contrast, permits – and even encourages – the assessment of each hit on the target as unique event which will, in each instance, yield one of a variety of randomly chosen – and often highly variant – outcomes. The deterministic or erosive type of model – in which Hood could never explode at all -- is well-suited to the comparison of tactical alternatives; in effect the ‘element of chance’ has been carefully edited out and each battle fought in identical circumstances will, or at least should, come out in exactly the same way. This characteristic enables users to test alternative ship designs in an objective manner – if the tactical situation begins the same, then changes in the result are inherently due to material changes in the characteristics of the firing ship, or the target. Conversely, if outcomes vary but ship characteristics etc. remain the same, then the difference must be due to variance in tactics. The erosive model only works some of the time; Lt. J. V. Chase, essentially the Thomas Edison of mathematical wargaming, himself recognized the problem with this ‘smoothed’ result, noting “...that the sudden destruction arising from any cause whatsoever” would tend to invalidate the analysis, especially if it took place near the beginning rather than near the end, of the engagement.

The stochastic model better-duplicates the inherently chaotic and unpredictable nature of real combat, but only at the cost of rendering the comparison of different tactical approaches more difficult, insofar as it blurs the effects of tactical variation and those resulting via pure chance. The choice need not be binary; better results can often be obtained by combining the two systems – essentially a adopting mathematical hybrid. The other alternative, equally good, is the one Santarini chooses; instead of adopting a hybrid model, he uses a purely determinative model in one place and a stochastic model in the other as the situation demands.

The first two chapters of this roughly 170 - page treatment provide a rather conventional overview of the strategic and operational background leading up to the action in Denmark Strait. While serious students of naval history may find much of this material redundant, readers new to the action will find them both readable and informative. More advanced readers will find the meat of the discussion beginning in Chapter 3, “Tactical Aspects”. This is, in practical terms, where the mathematical material begins.

The first part of the Santarini’s discussion revolves around an analysis of who held the upper hand at the beginning of the action. Although the outcome of the battle is bound to be somewhat stochastically determined, i.e. will be found to vary somewhat randomly from case to case, which side, if any, approached with an initial advantage, i.e. which side should have won most of the time? We have mathematical models to answer this sort of question extending back well over a century. Which model might we choose?

As the statistician George Box is reported to have said “Essentially, all models are wrong, but some are useful.” The first embryonic tactical models seem to date from early in the 20th Century. In the U.S. Navy Lt. J. V. Chase published the first equation sets describing combat attrition in 1902. Unfortunately (and unnecessarily) his work remained classified until the 1970s, depriving Chase of his primacy. The American Cdr. Bradley Fiske, who worked with Chase, published a variant of his equations in 1905. Apparently working independently, or nearly so, the Russian M. Osipov and the British Engineer Frederick Lanchester published nearly identical equations almost simultaneously in 1914-1915. In this case, RAdm. Santarini, probably correctly, chooses to use the “...simpler and more intuitive” 1905 method of Fiske over Lanchester’s “...more ‘elegant’ and rigorous mathematical approach” in his analysis. The results, in any case, are likely to be similar.

Thus armed for the fray, RAdm. Santarini elegantly wields a deterministic model to examine some 32 alternative scenarios, 16 initially favorable to the British, and 16 initially favorable to the Germans, concluding that the initial conditions of the action -- i.e. the conditions applying during the approach phases only -- gave the Germans only about 15% chance of victory.

A good deal, of course, depends upon exactly how one might define a ‘victory’; the Fiske model essentially equates this to complete annihilation of the enemy, not necessarily equivalent to what one might now define as a ‘mission kill’. Santarini concludes, as do most other researchers, that the British concern initially was primarily to keep ahead of the German formation, i.e. to cut the Germans off and prevent their entry into the Atlantic without having to participate in an engagement which might either sink or at the very least ‘mission-kill’ the German giant. This reviewer’s study of the Appolonios diagrams of the action suggests that, barring British ineptitude, the Germans could only have avoided an engagement by abandoning the attempt at a breakout entirely. Further, once apprised of their opposition, they would have realized at once that they were at a disadvantage – though perhaps not a large disadvantage – in almost any tactical situation. The main characteristics of the potential forthcoming engagement(s) – which, incidentally, would have been immediately apparent to tacticians on each side – fundamentally represented a variety of tradeoffs between one or more relatively brief short-ranged actions with high relative range and bearing rates vs a variety of relatively lengthy long-range actions with low relative range and bearing rates, in the extreme case defaulting into a stern-chase.

For a range of 15000 meters, once ‘on’ in spot, German tactical manuals would have suggested a 30% hit percentage on a battleship-sized target i.e. around 4.8 hits per minute, with Hood’s 15" guns returning on 3.2 hits per minute in return. Although the Germans lacked detailed information on the guns of POW, scaling 15" hitting power and rates of fire to 14" caliber would suggest that POW might have contributed approximately 3 additional 15" equivalent hits per minute, yielding an overall fire ratio of 4:8 German to 6.2 British, i.e about 1.3:1. Adding Prinz Eugen into the picture might increase the German firepower by 20%, reducing the ratio of equivalent 15" hits to about 5.75:6.2, i.e. about 1.08:1. Fiske’s attrition tables – which by their very nature preclude the possibility of a ‘critical hit’ such as the one which destroyed Hood – suggest a lengthy battle with the British certainly winning, at that point retaining slightly over 40% of their original offensive power.

Santarini and I agree that, all things considered, the Germans should have lost.

Chapter 4 – “Technical Analysis of KMS Bismarck’s Fire” represents a minor misnomer, as it actually includes a discussion of Prinz Eugen’s fire on the British as well. Noting some ambiguity in the cumulative attribution of hits on Prince of Wales and Hood, Santarini develops in detail two alternative hypotheses, one consisting of three 15" and four 8" hits, the other four 15" hits and three 8" hits. (As official data on the dispersion of the 20.3 SK S/34 guns of Prinz Eugen was not available to him during this phase of his study, Santarini uses alternative methods to assess these guns.) His conclusion is that Bismarck’s fire on Hood was not perfectly centered on the target until the forth or fifth salvo, which suggests, at least to him, the presence of systematic errors in the fall of shot of salvos two, three, and four. This is not surprising insofar as the first few salvos of an engagement are usually involved in ‘spotting on’, basically making corrections to fall of shot until a straddle occurs. Such‘systematic errors’ may spring from either defects in the mechanical fire-control solutions, poor choices in spotting – which in many navies was driven by doctrine – or both. Santarini’s analysis suggests that the problem was primarily mechanical.

Chapter 5 deals with Terminal ballistics. The treatment begins with a listing of a variety of hypotheses concerning the origin and nature of the blast which destroyed Hood, quoting Chesneau “Hood – Life and Death of a Battlecruiser” at length. Santarini calculates that during the action the total extent of Hood’s vertical surfaces (including underwater area) presented an area about 3.5 times greater than the total of her horizontal surfaces. (This reviewer calculates the ratio, taking into account only the area of the vitals, as roughly 2:1). This ratio, of course, enables the investigator to determine what percentage of hits might strike where. Santarini clearly knows just how far numbers can -- and can not -- take one; his somewhat simplified approach to armor penetration prediction is both refreshing and reasonable; instead of attempting to split mathematical and theoretical hairs, he cogently (and correctly) notes that “Ballistic penetration is a very complex phenomenon for which a well-defined scientific theory does not yet exist.” This justifies his subsequent use of a simple DeMarre multi-plate formula to compute the effective (rather than actual) deck thicknesses on Hood. These range between 96 and 131 mm, depending upon position fore and aft, taking the obliquity multiplier to be equal to 1/sin obliquity. At 63.2 degrees resolved obliquity, the effective thickness of Hood’s 305mm/178mm, and 127mm belts are computed to be 340mm, 200mm, and 140mm respectively. For a range of 15,200 meters, Santarini calculates the maximum penetration of belt armor to be 340-370mm, for deck armor possibly as much as 74mm, but probably more like 50-55, thereafter concluding that the most likely path of penetration would have involved a projectile which passed well above the belt, penetrated the main deck slightly to starboard of the centerline, and thereafter exploded in the centerline 4" ammunition working spaces on the main deck located between Frames 280 and 304. This is in complete accordance with previous findings. Unfortunately this is followed by a lengthy quote from Mearns and White suggesting that the separation of the wreck of the bow and the conning tower might have been due to the explosion of Hood’s forward magazines, an idea which has never been taken seriously by forensic experts. Careful analysis makes it clear that the bow separated due to hydrostatic implosive forces, and close examination of the wreck reveals that the conning tower simply fell off as the ship inverted on its course to the bottom.

The main thrust throughout this portion revolves around a rather conventional, though exquisitely detailed, statistical analysis of the gunnery action on a shell by shell and salvo by salvo basis. The accuracy and characteristics of both German and British fire are explored in almost equal detail. Some simplification is inevitable and appropriate; to keep the narrative going, readers interested in the details of technique are referred to one or more of six appendices.

Santarini contends that “...the destruction of a large warship by artillery fire alone can take place only if the hits affect ammunition magazines and the ammunition itself.” No argument there; most navies had early-on concluded that the number of hits required to actually sink a target was around two to four times the number required to effectively disable it, in some cases advocating switching targets earlier than might be intuitive, thereafter concentrating on the most dangerous remaining enemies and leaving the disabled to be dispatched at leisure after the main action had been completed. Assuming a hit has occurred, Santarini sees the outcome, at least so far as a magazine explosion is concerned, as the product of probabilities, in this case four, viz: the probability that a hit occurs in the right area of the ship, the probability that the bullet penetrates the protective armor, the probability that the projectile arrives in a position where its detonation would ignite the magazine(s), and the probability that the fuze would function correctly. Taking the probability of intercepting either the fore or aft magazine group as equally likely, assuming perfectly ‘on-target’ firing and making ‘reasonable estimates’ of the probability of each component, he concludes the Single Shot Hit Probabilities (SSHP) on Hood for Bismarck’s 2nd, 3rd, 4th, and 5th salvo to be approximately equal to 0.075, 0.109, 0.116, and 0.125 respectively, and the probability of a kill (Pk) to be 4.8% for a hit occurring at any time during the engagement. The author calculates that a 95% chance of killing Hood under the circumstances existing during Bismarck’s fifth salvo would have required the expenditure of about 320 rounds of ammunition, and therefore that Hood’s demise after 32 rounds, “...is therefore to be regarded as a highly improbable event.” Although others might structure Santarini’s probability cascade somewhat differently, and assume different quotients, the overall outcome likely to be similar.

The final chapter – “A New Perspective” – is devoted to the rejection of a few popular myths. First, the author concludes that German gunnery was not remarkably good, noting that “there is about an 85% probability that the Bismarck’s firing at the Hood was affected by a substantial systematic error”, acknowledging that this problem seems to have disappeared or been worked out by the time Bismarck engaged Prince of Wales. In spite of the fact that the British took more than five salvos to first cross the target – a sequence which most navies would have seen as intolerable – Santarini’s conclusion is that British gunnery was good. Prince of Wales, he says “...fired most effectively between 0553 and 0600 scoring – even under unfavorable conditions – as many as three hits against the Bismarck. The restricted dispersion of the British guns, the excellent alignment of the heavy-calibre battery, and above all, the outstanding ability of the first gunnery officer...account for this very good performance.”

It is refreshing to note that the author, in the company of all other serious researchers into this action, clearly rejects the hypothesis that ‘plunging fire’ from Prinz Eugen could have caused the fatal blow. So far as Hood goes, his conclusion is that “Reconstructions indicate that the explosion took placed in the vicinity of the 341/42 aft magazines. Actually, bursting in the AA 102/45 ammunition working spaces located below the upper deck by the X 381/42 turret seems much more realistic. The detonation of the AA medium-calibre projectiles would then propagate in the propelling charges magazine.” Although the protection of Hood was clearly not up to the task – barring truly unusual circumstances the ship would not have been lost otherwise – the author assigns the explosion more to bad luck than poor design, observing that the assignment by some historians of almost magical properties to Bismarck’s gunnery and projectiles represents, in his words, an “over-exaggeration”. This may be true now, but not necessarily then, there is some possibility that one of the reasons Prince of Wales discontinued the action revolved around concerns that the sudden loss of Hood suggested that Germans might have developed some sort of particularly lethal projectile design. Examination of a ‘dud’ removed from Prince of Wales some time after the action revealed that German projectile design was, in reality, fairly mundane.

As noted earlier, the author has wisely relegated the most rigorous mathematics to six appendicies at the end of the book, totaling 41 pages. The first four of these explain in detail the derivation of target angles and ranges, the calculation of hitting space and Single Shot Hit Probabilities, the dispersion characteristics of Bismarck’s guns, and a statistical analysis of Bismarck’s firing accuracy. Non-mathematicians may find this weighty stuff – although the statistical techniques employed by the author are relatively simple, many readers -- particularly those who are more accustomed to reading historical narrative -- may find much of the mathematics both intimidating and unfamiliar. Fortunately, complete comprehension of the mathematical processes involved represents no prerequisite to understanding the conclusions presented. The last two appendices contain specifications of Bismarck’s guns and a brief discussion Lanchester’s “Laws of Concentration’. The appendices are, in turn, followed by 16 pages of endnotes.

The book includes 16 pages of colored illustrations, accompanied by a number of black and white photographs and diagrams which enable the reader to both understand the action and follow the statistical arguments quite clearly. The three-page bibliography, which does not list every source used by the author, primarily consists of secondary sources; this reader found some of the lesser-known Italian ones both interesting and informative. The reference to Antonio Bonomi’s recent (and ongoing) work on the tactical geometries of the Denmark Strait action is heartening; treating Mearns and White as a serious reference source – sometimes quoted at length – somewhat less so.

The text overall is written in fluent and readable English. Although the book does contain a few minor typographical errors, none of these might be considered critical. There is no index.

Those who with Santarini’s findings might characterize them as just one more example of what can be achieved while using “Lies, Damn Lies, and Statistics”. (As the ill-fated Frank Marble once wrote “A mathematical deduction has no more validity than the premises upon which it is founded; and a series of approximations or guesses does not become true because it is expressed in algebraical terms.”) In this case they would be wrong. There need be no fear of mathematical chicanery here. Santarini’s thorough and professional analysis represents both a welcome and useful addition to the literature on the Battle of Denmark Strait.

Bill Jurens

Bismarck and Hood

The Battle of the Denmark Strait – A Technical Analysis For A New Perspective

by RAdm. Marco Santarini

Published by Fonthill Media Limited

ISBN: 07522978-1-78155-231-5, Published 2013

Approximate Retail Price: £20

It is available through many local book retailers as well as online sellers such as Amazon.com and BOL