Bootstrapping

In their paper "Bootstrapping the Battle of Britain" which was published in The Journal of Military History (84(1), 151-186), Fagan, BT, Horwood, I, MacKay, NJ, Price, C, Richards, E & Wood, AJ discussed using the statistical technique of weighted bootstrapping to model a hypothesis about the battle. As I have my own data set I have explored using boostrap simulations to investigate aspects of the battle.

What is Boostrapping?

Bootstrapping uses sampling with replacement to generate an alternative data set based on the original. For example, if we have 30 data points then to create an equivalent bootstrap sample we would randomly select one of those data points 30 times, picking from the full data set each time meaning each original data point could appear more than once (or not at all) in the bootstrap sample. This is what's meant by "replacement", each time we pick a data point we place it back into the sample pool we are drawing from. By running this sampling process a large number of times the aggregate bootstrap data sets should be representative of the original data set.

Block boostrapping

The Battle of Britain is typically broken up into phases (e.g. Kanalcampf, Adlerangriff, the Blitz) so for bootstrapping the whole data set can be broken down into sub-sets that correspond to these phases. For example, if we define Adlerangriff as lasting from 12 August to 17 September then we can create bootstrap data sets from this block to represent a hypothetical Adlerangriff.

Re-weighting

Using blocks enables the creation of "what if?" scenarios where we model what would happen if Adlerangriff had been started earlier and run for longer by creating bootstrap data sets that are weighted differently to the original.

Modelling

For my modelling I have taken the approach of using a starting value (S0) and walking through the data set applying the value for each day (Lt) to create a simulated run of the battle.

The value on each day is thus: St = S(t-1) + Lt

Backtesting

In order to test the validity of the model we need to run it against the historical data to check it at least approximates the historical results, if it doesn't resemble the historical data then it suggests that there's something wrong with our model or assumptions.

For example if we look at the classic example of R.A.F. fighter pilot strength we can use the historical values for this during the battle given in Appendix 11 of The Narrow Margin:

Date Strength
1940-07-06 1,259
1940-07-13 1,341
1940-07-20 1,365
1940-07-27 1,377
1940-08-03 1,434
1940-08-10 1,396
1940-08-17 1,379
1940-08-24 1,377
1940-08-31 1,422

To run our model we start with the value for S0 of 1,341 on 13 July 1940 (as that's the first day that is in both our data set and the historical values) then for each day in our data set we subtract the number of pilots lost or wounded (Lt) and add on the number of replacement pilots per day that Fighter Command was receiving (r) this gives us a figure for the R.A.F.'s pilot strength on each day:

St = S(t-1) - Lt + r

Before creating our bootstrapped data sets we run this on our data set in chronological order. As r is not a figure for which I have a source, runs were done with different values to establish a best fit.

Bootstrappd R.A.F. pilot strength (simulated with replacement rate)

As the shape of these graphs matches the recorded pilot numbers it indicates that the model is valid and a value for r of six provides a reasonable fit for this data set.

This value of r is not to suggest that this was the actual pilot replacement value but merely a value that is required to make our model "work" - it's a fudge factor.

(There is a notable divergence for 31 August - notably one of the days where R.A.F. single day losses were very high - but, as that's the limit of my data set, I'm ignoring it for now. That is something that will need more data to investigate but my initial guess is that the value for r will need to change at this point if the model is to follow the historical data.)

Testing

Now we have some confidence in our model it's time to put the bootstrap to work in order to investigate some aspects of the battle.

Initially it's helpful to understand what the bootstrap simulation does. For the backtest we used the values of Lt in chronological order. For the boostrapped model we create new, randomly selected, sequences of days to run through the model and then then average (rounded mean) the values for St from each day of all the runs to give us a sequence of values. Unless otherwise stated, the simulations here are using 10,000 runs.

If we treat the period we're examining as a single phase the boostrapped result gives us a straight line between the starting strength and the ending strength. This intuitively makes sense as we're taking random samples of each day and averaging them out so we would expect it to produce a straight line average across all days.

Bootstrappd R.A.F. pilot strength (single phase)

I take two things away from this: The first is that viewing this data as a single phase isn't particularly helpful as the simulated result isn't a good match for the shape of historical data. The second is that picking the start and end point of phases is going to be very important as the result of that phase will be the average of the start and end points.

Phasing

The Battle of Britain is typically viewed as series of distinct phases with those established in 1943 by the official R.A.F. history of the battle being the most common definition.

Dates Phase
1940-07-13 - 1940-08-07 Phase I: Channel Battles
1940-08-08 - 1940-08-18 Phase II
1940-08-19 - 1940-08-23 Pause
1940-08-24 - 1940-09-06 Phase III
1940-09-07 - 1940-10-21 Phase IV

Running a bootstrap using these phases gives a good fit for the model.

Bootstrappd R.A.F. pilot strength (R.A.F. phase asignment)

Whilst the R.A.F. phases match up well with R.A.F. pilot numbers they don't match up with the phases of strategy employed by the Luftwaffe which are outlined as follows in "To Defeat the Few":

Dates Phase
1940-07-13 - 1940-08-10 Kanalcampf
1940-08-12 - 1940-08-16 Adlerangriff stage 1
1940-08-18 - 1940-08-22 Adlerangriff stage 2
1940-08-24 - 1940-09-06 Adlerangriff stage 3
1940-09-07 - 1940-09-17 Adlerangriff stage 4

This ends on 17 September as this was when Seelöwe was officially cancelled so, whilst fighting continued, it marks the end of the Luftwaffe's attempt to obtain air superiority for an invasion.

Notably the following dates are not assigned to a phase: 1940-08-11, 1940-08-17 & 1940-08-23. For the purposes of modelling these have been assigned to a "Pause" phase which is used for these three dates in the sequence.

Running a bootstrap for these phases doesn't give as good a fit as the R.A.F. phases - although it's still in the ballpark of the historically recorded figures.

Bootstrappd R.A.F. pilot strength (Luftwaffe phase asignment)

Digging deeper into this there is one day that stands out as having a disproportionate effect on the result: 11 August. This is not assigned to a phase but has a high number of pilot losses (Lt) and looking at the events of that day there was a large attack from Luftflotten 3 on Portland docks so, in terms of tactics employed, it's probably a better fit for Kanalcampf. Running a bootstrap for this revised phasing gives a better fit for the first and third stages of Adlerangriff.

Bootstrappd R.A.F. pilot strength (Revised Kanalcampf asignment)

There is still one outlier here and that's 18th August which, with attacks on satellite airfields, is more closely aligned to stage two of Adlerangriff. Assigning the two remaining "pause" phases to the nearest applicable stage gives us a phasing that is similar to the R.A.F. assignment.

Dates Phase
1940-07-13 - 1940-08-11 Kanalcampf
1940-08-12 - 1940-08-18 Adlerangriff stage 1
1940-08-19 - 1940-08-23 Adlerangriff stage 2
1940-08-24 - 1940-09-06 Adlerangriff stage 3
1940-09-07 - 1940-09-17 Adlerangriff stage 4

Running a bootstrap for this gives a good fit for the stages of Luftwaffe operations whilst matching their strategic phasing.

Bootstrappd R.A.F. pilot strength (Fitted Adlerangriff phase asignment)

This now gives us a reasonable basis for examining alternate scenarios of Luftwaffe strategy.

EprGr 210 attack four radar stations

Visibility is perfect with Dungeness being visible from Dover.

The Bf 110s of EprGr 210 lead by Swiss born Hptm Rübensdörffer come in at low level to attack the radar network. Lacking azimuth tracking and confusing range information by passing abeam the radar sites, the Filter Room at Fighter Command assigns the plot an 'X' code (doubtful origin).

Oberleutnant Otto Hintze leads 3 Staffel's eight Bf 109E-1/B Jabos - carrying one SC250 bomb each - attacking Dover. They approach the Dover radar at right angles to make them harder to detect but are picked up by the Rye Radar. The towers are slightly damaged and huts are smashed. Hintze reports seeing the towers clearly swaying.

Two minutes later Lieutenant Rössinger leads 2 Staffel's Bf 110s attacking Rye. Almost every building is hit, except the transmitting and receiving block. The Filter Room repeatedly tries to contact them eventually being told by a WAFF that "Your X raid is bombing us."

Hauptman Rübensdörffer leads the attack on Dunkirk. One 500kg bomb moves the concrete transmitter building several inches.

The final element of 1 Staffel's Bf 110s hit Pevensy with eight 500kg bombs, one of which cuts the power line taking the station off line.

A breach of 160 km has been opened in the radar coverage.

151 Sqn and 501 Sqn attack IV StG 1 off Margate

15 Hurricanes attack 22 Ju 87s returning after attacking convoys Arena and Agent north of Margate. One Stuka is shot down before Adolf Galland's III/JG 26 arrive shooting down two Hurricanes and damaging two more.

Margate motor lifeboat J. B. Proudfoot is launched to pick up the survivors of Admiralty trawlers Pyrope and Tamarisk.

KG 51 bomb Ventnor radar and Portsmouth harbour

Escorted by 120 Bf 110s of III/ZG 76 and 25 Bf 109s of JG 53 as top cover are picked up by Polling radar at noon: a large formation approaching Brighton from due south before crossing the coast they turn west. As they cross in front of Polling and behind Ventor they are tracked by the ROC, splitting as they come to Spithead.

15 specially selected crews (II/KG 51) dive bomb droping 500kg bombs on Ventnor radar station taking it out of action for three days.  15 of the 74 bombs land in the complex.

A second formation of 70(68?) Ju 88s attacks Portsmouth harbour with horizontal and diving bomb runs in the face of 50 barrage balloons and intensive AA fire from guns on shore and ships. The bombers take advantage of the corridor through the barrage balloons at the harbour entrance.

The Hurricanes and Spitfires of 10 Group (152 , 213 & 609 Sqns) scrambled to intercept are vectored in by controllers in small groups and focus on the bombers whilst their escort, circling at 3,000m, wait for a large fighter group to arrive. As more and more bombers are shot down the escort realise that no large formation is going to present itself as a target and engages in the combat.

In the ensuing melee - that involved nearly 300 fighters in total - 13 RAF fighters are shot down and four are damaged, and five Bf 110s and two Bf 109s are lost.

More Bf 109s - the escort for the return leg - are seen heading west but Fighter Command vectors 615 Sqn's Hurricanes between them and the battle over the Isle of Wight.

12(9?) Do 17s were lost including that of the Geschwaderkommodore Oberst Dr Johnan-Volkmar Fisser who lead the attack on Ventnor.

Later German reconnaissance of Ventnor reports craters in the vicinity of the masts and station quarters on fire.

KG 76 bomb Hawkinge

At Hawkinge the personnel on the base are taken by surprise as the Do 17s (KG 2 and KG 76) and Bf 110s (EprGr 210) come in over Folkestone at medium height and the AA crews aren't able to man their guns - with the exception of two Hisapnios.

Number three hanger at Hawkinge receives direct hits and the iron doors come off their runners killing one airman and two civilian employees - Brisley and McCaister. Inside the hanger two Spitfires under repair were wrecked and two others seriously damaged. Four airmen are killed and six seriously wounded.

The workshops, clothing store and two married quarters were destroyed and the main store was on fire and the ammunition there was exploding.

Luftwaffe records state that Hawkinge was attacked in two waves, with 280 110-lb bombs being dropped first, followed by 16 1100-lb and eight 551-lb bombs.

64 Sqn's Spitfres engaged the Do 17s over Hawkinge and remarked on the lack of AA fire. American PO Donahue bailed out of his burning Spitfire over Sellinge.

Hurricanes from 32 Sqn returning to the airfield to refuel have to negotiate the 28 craters on the landing field. Flt. Lt. Michael Crossley requests permission to land, the response from the controller being "Hello Jacko Red Leader. We've had a spot of bother here. Permission to pancake granted. Good Luck." Sgt. Lacey's undercarriage collapses and F/Lt Gibson's aircraft ends up on it's nose. PO Barton opts to crash-land in a nearby field. The aircraft are refuelled and returned to Biggin Hill.

Firemen from Folkestone arrive to help out at the airfield. Section Leader R. R. Fry is accompanied by his team of Bill Willis, Percy Sutton, Ted Beeching, George Rumsey and George Kelly are fighting the multiple blazes at No 3 hanger where saving the aircraft isn't possible.

49 MU lost a number of vehicels, including a Ford V8 staff car and a Commer low-loader.

Fry said of the incident:

A chaotic scene greeted our arrival. There were fires in several places. The water tower, supplying pressure for the hydrants, had been holed by splinters in may places and water cascaded from it, reducing our mains supply to a trickle as we watched.

The main priority was getting the landing field operational and soldiers, airmen and civilians worked through the night filling in craters, sustained by a mobile canteen serving tea and sandwiches.

OPERATIONS RECORD BOOK of (Unit or Formation) RAF. Station Hawkinge

Place Date Time Summary of Events References to Appendices
Hawkinge August 1940
12th
WEATHER. fine. The I.G. of the R.A.F, Sir E.A LUDLOW-HEWITT, K.C.B., C.M.G., D.S.O., M.C. inspected the Station. An Enemy Bombing attack by J.U. 88's against the Station was carried out at 17.30 hours and lasted for approximately 10 minutes. One hanger, No. 3, was almost completely wrecked whilst on other, No. 5, was partially wrecked. A number of Bombs of heavy caliber, including incendiary, were dropped. The aerodrome and buildings were machine-gunned during the attack. The main stores were partly damaged by fire, the clothing store almost completely. The fire was quickly brought under control by R.A.F. personnel aided by local A.F.S. The Station Workshops were wrecked. Two houses in the Airmen's Married Quarters , occupied by airmen, were destroyed. Twenty-eight craters were made on the aerodrome, the largest being 76' x 72' x 28' deep, and the smallest 10' x 10' x 8' deep, but the aerodrome was not rendered completely unserviceable. Repairs to the surface were immediately commenced by the R.E.s. already attached for such work. Ground Defenses were surprised and no guns, except two HISPANO were fired. The altitude of the attacking aircraft was such that it was impractical for the P.A.C.U. to be brought into action. Two civilians, MR. BRISLEY and MR. CAISTER, employed by contractors of the Works Directorate, were killed and three airmen, CORPORAL Mc. COLL attached from R.A.F. Station, YEADON, A.C.2. SYMES attached from R.A.F. Station, KENLEY, and A.C.1. LANGDON of R.A.F. Station, HAWKINGE, were killed. Six airmen received severe injuries and were admitted to the KENT and CANTERBURY HOSPITAL, CANTERBURY. TheCasulaties occurred to personnel employed in No 3 hanger. Two SPITFIRE aircraft, under repair, were seriously damaged, whilst one or two others were struck by splinters. The two non-operational aircraft on charge were damaged but repairable. P.O. N.G. DURHAM reporting on posting from R.A.F. Station, WICK for duty at A.M.E.S., DOVER. F.O. J.D. GABB and F.O. J.H. READ reported on posting for operations room duties.
AIR RAID WARNINGS.
RED 08.35, WHITE 09.25, RED 11.38, WHITE 12.37, RED 23.37, WHITE 23.59

The identification of the attackers as Ju 88s seems incorrect.

German intelligence summary

German intelligence reports 71 aircraft destroyed - including all of 65 Sqn on the ground at Manston. (Actual RAF losses were 22 from 732 sorties.)

Luftwaffe signals chief Martini assesses that all radar stations are still operational. (Ventnor appeared on line as it is transmitting powered by a backup generator but it is unable to receive any echoes so is actually not functioning.) Conclusion is that operations rooms must be well protected underground and it is only possible to shut down radar coverage for short periods.

Reichsmarscahll Göring gives the order to launch Adler Tag the following day - this is intercepted by the British Ultra.

KG2 take off from Cambrai and Saint-Léger-lès-Authie

KG 54 and JG 2 cross the channel

Radar picks up a formation approaching the coast between Hastings and Bognor. 11 Group scrambles four squadrons with 43 Sqn making first contact with a fighter sweep of Bf 109s and Bf 110s of JG 2. As the fight moves westward more Bf 109s join and RAF fighters return to base with their ammunition expended. 238 Sqn (10 Group, Hurricanes) climb to meet the incoming fighters.

KG 54's Ju 88s are behind the fighter sweep aiming for Odiham and RAE Farnborough and they are spotted by 601 Sqn's A flight, who were climbing to meet the fighter sweep. 43 Sqn, 601 Sqn and 64 Sqn all join the attack. Unable to navigate due to the cloud cover and attacked by fighters they turn back 10 miles after crossing the coast.

KG 2 intercepted

Due to the heavy cloud the Observer Corps are unable to get a visual fix on the aircraft and the number of raiders is incorrectly assessed so of the five squadrons scrambled to intercept only 74 Sqn manage to engage them before they reach the Isle of Sheppy and split into two groups: one to attack Sheerness (a Naval base) and the other Eastchurch (a Coastal Command airfield).

Göring addresses Luftflotten commanders

At Karinhall Göring says to the commanders of his Luftflotten:

The fighter escort defences of out Stuka formations must be readjusted as the enemy is concentrating his fighters against our Stuka operations. It appears necessary to allocate three fighter Gruppen to each Stuka Gruppe, on of these fighter Gruppen remains with the Skukas and dives with themto the attack; the second files ahead of the target at medium altitude and engages fighter defecnesl the third protects the whole attack from above. It will also be necessary to escort Stukas returning from the attack over the Channel.

We must concentrate our attacks on the destruction of the enemy air forces.
Operations are to be directed exclusively against the enemy air force including the targets of the enemy aircraft industry... Our night attacks are essentially dislocation raids, made so the enemy defences and population shall be allowed no respite.

It is doubtful weather there is any point in continuing the attacks on radar sites, in view of the fact that none of those attacked so far have neem put out of action.

Later in the day he issues an order prohibiting more than one officer in any single air crew in an attempt to reduce officer casualties on bombers which are reaching serious proportions.

Kesselring decides to use heavy escorted raids on fewer, selected airfields, along with fighter missions.