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Computer Simulation Games - Modelling WWII Dive Bombers

Paul F. Watson
January 2020

Introduction: I have always wondered how accurate those WWII dive bombers were. How often did they hit a ship? Could they hit a tank? These kinds of questions led me to perform a careful statistical analysis and to summarise the results in the article below. For general information about dive bombers, I recommend reading Peter Smith’s excellent book Dive Bomber!. What I offer in this article are conclusions based on statistical analysis -- an area not routinely trod by historians; but, I gladly acknowledge the hard work of historians like Peter who made this analysis possible. I would also like to recommend that readers interested in the development and history of dive bombers should read Peter C. Smith's book Dive Bomber!. It is outstanding.

Intended Use of Results: The following analysis conclusions are presented to enable realistic simulation game modelling of WWII dive bomber attacks. A primary effort has been placed on analysing Pacific War dive bomber attacks against ships, but a reasonable effort to characterise interwar British testing of land based dive bombers is also included. Casual use of Gaussian statistics was used for the land based analysis and Weibull Analysis was used for the Pacific War analysis. A comparison of the land vs. naval results indicate roughly the same average accuracy.

The emphasis of this research was to determine ‘where did the bomb fall’, or ‘what percent of bombs dropped during a saturation attack likely struck a ship.’ This research is not intended to determine the destructive effects of such bomb drops. A few comments on the subject are presented in Appendix B.

Credits: Use of the models presented herein are allowed (without payment) provided credit is provided in introductory/lead-in material for games along with a link to www.dionysus.biz. Credit should be styled as per Appendix C Credits, or as authorised by me at paul-watson@sbcglobal.net

I wish to acknowledge the scholarship and readability of the very interesting book by Peter Smith entitled Dive Bomber!. Peter’s book provided both broad understanding of the subject and also provided most of the historical data concerning interwar testing and numerous instances of attack success against shipping in the Pacific. See Sources for this and other sources that served as a basis for this article.

Method of Analysis: Both Gaussian statistics and Weibull Analysis were used in preparing the following summary conclusions. The conclusions presented herein are based on sound technical methods of analysis used in science and industry. I may, at a later date present a detailed article on how I analyzed dive bomber accuracy. I might add that this paper proved to be a rather tedious undertaking which is the reason I am not jumping into a technical explanation. The conclusions presented herein are about as thorough and accurate as can be made looking through the gloom of time to events almost a century ago.

Early British Dive Bomber History (1918 to 1940): Numerous WWI land based attacks were carried out with bomb release during a dive. Such attacks seemed more accurate than level bombing attacks. By 1918, the British were conducting land based tests on dive bombing methods. Stationary target accuracy tests were performed using five pilots and SE5A aircraft. The primary experimental variable was bomb release height. See Table 1 for results. (1)

Between 1918 and 1938, numerous additional experiments were conducted but data correlation between experiments was difficult because no standard dive bombing doctrine existed which would have facilitated orderly test deviation from ‘controlled conditions’. Two studies that do seem to offer similar data are presented in Table 1 below. Both tests, examined dependency of target error on release altitude. (1)

Information from Table 1 can be reduced to a graph showing the effect of bomb release altitude on radial error (i.e. distance from center of target in feet). See Figure 1 below:

British Pre-WWII Dive Bomber Radial Accuracy in Feet
Figure 1

Figure 1 demonstrates that lower altitude bomb releases results in less target error. If this trend is reduced to a straight line, it would be the blue line shown in Figure 1. For programmers, it is useful to describe the blue line by Equation 1 below.

• Eq 1: Radial target error in feet = .061 * Release Height in feet – 1.584
• The R2 value of 83% indicates a ‘ratty’ fit line, that is better than guessing.

We might summarise Equation 1 by saying that the typical (I believe average) bomb distance from center of target is about 6% of bomb release altitude. Thus, lower drop height means greater accuracy! It also means greater risk from anti aircraft fire, and ground impact.

• Dive bombing is far more accurate than level bombing
• Steeper dives improve accuracy
• Lower bomb release improves accuracy. In reference 6 Stuka Pilot, German pilot Rudel confirms this, stating his preference for lower bomb release to improve accuracy.

Method of Analysis- British Land Based Dive Bomber Accuracy: Statistical analysis always assumes something about the underlying data. In general, we prefer to analyse data belonging to a single population. For Dive Bombers, we can assume differences exist in pilot training and aircraft between British, Americans, Japanese, Germans, Italians etc.. Naval pilot training likely differs from army air corps training. We should thus be cautious in assuming similar methods and results across organisational boundaries.

Conclusions for British Land Based & Comparison to Navy & German: Clear answers about dive bomber accuracy become entangled with the attack doctrine of various services. My analysis assumes the following for all WWII dive bombers:

• A single statistic called ‘radial target error’ in feet is a legitimate measure of accuracy.
• Bomb releases were at 1500 feet altitude for dive bombers of all services and nations.

Table 2 provides a comparison of British, German (unreferenced data from Internet forum), Japanese Navy, and US Navy based on Watson Gaussian Analysis. A discussion explaining the significance of the data follows the table. The U.S. Navy Avg.Rad.Err.ft value has been worked out here is the example.. The www.dionysus.biz home page also has an introductory statistics course available for free, on line. Go to www.dionysus.biz and follow the link to St Pauls Introduction to Statistics if interested.

Comments:

1. British land based testing was conducted under peacetime conditions, without anti-aircraft fire and likely short mission duration. The very favourable average radial miss distance is likely optimistic; yet, results ‘hovering’ about 100 foot radial error for many table entries suggests results of Table 2 are about right.
2. German data was obtained from a non-foot noted Internet forum with source and data conditions undocumented. This source indicates about double the target 'miss' of allied pilots; and yet, considerable evidence suggests otherwise. Peter Smith indicates that Stukas were the one land based dive bomber that was effective against ships. This suggests Stuka accuracy was similar to to US and Japanese Naval dive bomber performance. In Stuka Pilot (see reference 6), Hans Rudel indicates that during practice he rarely went wide of the target by 30 feet once again casting doubt on the average miss value given in my Table 2. Rudel recounts a large number of tanks, that he personally destroyed (many with a 37mm airborne cannon). While he was a remarkably skilled pilot, these statements call to question an average radial error of 200 feet.
3. Table 2 naval attack data for both US and Japan is derived from detailed Gaussian statistical analysis of attack report summaries from Peter Smith’s book (see Appendices). For Gaussian analysis of Naval accuracy, a target beam/width of 63 feet (heavy cruiser) was assumed.

Discussion of Table 2: Naval records make clear that dive bomber hit percentages were highly variable from one mission to the next. Some missions simply missed the target, while others achieved 50% hits. This variability was likely typical for land based dive bombers as well. During WWII, there was no “standard hit percentage” that could be counted on; but long term averages can be established and we can characterise the variability between missions.

The average radial error in feet describes how many feet away from the centre of a target a bomb typically fell (with some falling closer and some farther away). The radial error numbers of Table 2 generally indicate dive bombers could hit, on average within 100 feet of their target.

The Standard Deviation provided in Table 2 characterises how variable mission results were. Mathematicians interested analysing WWII dive bomber attacks will need the averages and the standard deviation (both are provided in Table 2). This will allow calculation of the likelihood of a bomb hitting at approximate distances from the target.

Analysis of WWII Naval Dive-bomber accuracy: Some discussion of both U.S. and Japanese naval dive bomber accuracy has already been presented. The following section provides more complete information including:

• Analysis goals
• Analysis method
• Analysis of mission success variability & its use in computer simulation games.

Realistic Simulation Games strive to provide ‘game players’ with a realistic simulation of the ‘ups and downs’ of the historical experience, minus the bloodshed. Dive bomber performance is made up of two statistical components: How accurate are the bombs? (i.e. What percentage are likely to hit a target ship?) And what is the mission to mission variation that can be expected?

1. The first question (how accurate) has largely been answered by Table 2.
2. When discussing the WWII Pacific theatre, attacks were typically ‘saturation attacks’ carried out by 12 to 40 aircraft, attacking in a very short period of time. In such circumstance, we may assume that bomb hits will occur, but the success of attacks is highly variable. In the paragraphs that follow, we characterise the relative likelihood of 10% hits or 25% hits or 35% hits etc.

Interwar US Navy Dive Bomber Research: October 1926, LCDR Frank Wagner carried out a mock attack on U.S. Pacific Fleet ships. Although the ships were notified in advance, Wagner's dive from 12000 feet caught the fleet with “guns unmanned”. (Ref Globe Security.Org. Author not identified. See also Smith).

Wagner later demonstrated dive bomber accuracy tests scoring 19 hits out of 45 bombs aimed at a 100'x45' target. (Ref Globe Security.Org). Note: if target short axis is assumed to dominate targeting errors and data is analyzed by Standard Normal Distribution, this would indicate a Standard Deviation of 36.8 feet.

Circa 1928: Lieutenant Commander Leighton gave a lecture at the Naval War College & indicated the following (See Smith, ‘DiveBomber!’):

1. Dive bombers are little vulnerable to antiaircraft fire (level bombers are vulnerable)
2. Typical bomb release was 1500 foot altitude or less
3. Target practice on towed 20 kt. targets indicated 30% hits on 4 destroyer target. 50% on light cruiser targets & 60% on battleship targets. War experience proved these numbers optimistic.

In the Late 1930's, Captain McGee U.S.M.C. issued an ACTS Research Paper that summarised standard U.S. Naval tactics.

1. Arrival at target area. Push-over into steep dive from 8000 foot altitude. Bomb release at 2000-3000 feet with recovery about 1000 feet. Attacks were typically at 5 to 10 second intervals and varied direction of approach.
2. Dive bombing reduced radial error of horizontal bombing against stationary targets by at least 50%. Against moving targets, the improvement was greater.
3. TBD source indicates U.S. doctrine called for a 70 degree dive angle whereas Japanese practice called for 50 degree dive angle. Source also indicates 1500 foot US Bomb release.

WWII Pacific Dive Bomber Attacks: Experience & Interpretations: Analyses of wartime data indicates McGee's dive-bomber hit percentages were about right for cruisers, but optimistic regarding battleship hits. For level bombing, McGee’s information is unrealistically optimistic. (History records a single instance of a ship (destroyer) hit by level bombing). The following pages present a statistical analysis of WWII combat data extracted from Peter Smith's book. The statistical analysis resulted in two equations (one for US/UK and a second for Japan) that describe mission to mission success variability. These have been reduced to “dice hit tables” as a simple way of conveying the conclusions. Game developers may prefer to use random number generators and equations. An example of these methods is also presented.

Method of Analysis: The very powerful Weibull Statistics method was used to characterise variability in mission strike effectiveness. This article presents results, rather than method; but a brief summary seems appropriate. Weibull Analysis is performed as follows:

1. The data is collected & organised in a form suitable for analysis.
2. A Weibull Plot, like Figure 2 is used to determine whether Weibull describes the data.
3. If Weibull is good, Beta & Eta numbers from step 2 are used to write PDF equations describing variability
4. From the PDF equation of step 3, data is processed in a spreadsheet to extract whatever information we desire (e.g. such as mission to mission variability).

Because the focus of this article is computer simulation games, I will leverage off the history of such games and their use of dice. I will present the statistical conclusions regarding mission variability as a “percent hit table” that correlates to a dice role. Data will be presented for 6 sided dice, equivalent data for 12 sided dice and equivalent data for 20 sided dice. The approach for game programmers using random number generators and equations is also presented below.

Weibull Plot - US Dive Bomber Attack Percent Hits
Figure 2

Figure 2 above is a Weibull plot based on 12 US Dive-bomber saturation attacks and their corresponding percent hits. The bottom and right axes are important (ignore top & left axes). The Weibull analysis program is currently under development and updates will allow me to label the x axis as “Mission Percent Hits”. A similar Weibull plot was prepared for Japanese air strikes.

Two things are important about Figure 2. First, the data points fall very close to a straight line. That means that Weibull will accurately model the data. Second, the Beta and Eta values shown at top left are used to write the Weibull equations that can be plugged into a spreadsheet and mined to determine how a dice role should correspond to percent hits achieved during an attack. These methods realistically describe both the percent hits and the “ups and downs” experience of a naval commander.

Dice Hit Tables from Weibull Analysis:The following dice hit tables were created from Weibull results of separate analyses for Japanese and U.S. attack data. A possible future article may explain the analysis method. The results are presented below for whatever dice you may wish to use.

Converting Dice Tables to Usable Equations for Computers: It is not difficult to reduce one of these tables to an equivalent equation form (You will need 1 equation for US and a second equation for Japanese). For simplicity, I used U.S. data from Table 4 to create an example. See also Figures 3 & 4 below:

1. Using a Spreadsheet: Enter dice role in column A. Enter hit percentage in column B.
2. Use spreadsheet to graph data. Select x-y graphing option as shown in Figure 3 below.
3. Choose 'trend-line' option for your graph. You may have to 'click' a data point for a 'pop-up' command box. Select 'display equation' option
4. Tables can be used with dice. For computers, a random number generator with equations are better. An exponential equation worked best for 6 sided dice table:

Screen capture of Steps 1 & 2 Shown
Figure 3


Screen showing 'Trend Line' and Trend Line Equation
Figure 4

Program Function or Subroutine: The game programmer can use code something like this in function or subroutine call (pseudo code follows):

	DefFunction USHitPercent
	DieRole=Random(Int(6))
	If DieRole>5:
              HitPrcnt=0.00
	Else: HitPrcnt=.04875*(2.71828^(.410018*DieRole) #REM Equation copied from Figure 4 trend-line
	Return HitPrcnt
	

Width Correction Factor: The percent hit tables above are based on typical size of a ship. For game modellers, a linear correction factor is recommended as follows:

• Corrected Hit Percent = [(Actual Target Beam in feet)/63] * Table Hit Percent

Conclusions: Mathematical and statistical analysis has been used to accurately describe WWII dive bomber performance. The following information has been provided:

• Average ‘miss distance in feet’ for land and sea based dive bombers (Table 2)
• Statistical value Standard Deviation provided to allow Gaussian analysis (Table 2)
• Documented model of bomb release altitude affect on target accuracy (Figure 1)
• Interwar US Naval summaries of predicted percent hits on naval targets (Table 3)
• For the U.S., long term naval dive bomber percent hits for experienced pilots was 16%.
• For Japan, long term naval dive bomber hits was 18.2% from 1940-1943. After 1943, hit rate plummeted.
• Reduction of Weibull Analysis on naval strike variability into ‘dice probability’ tables that predict percent hits for a specific mission (see Tables 4, 5 & 6)
• Example of how dice tables 4, 5, & 6 can be used within computer programs.
• Target beam width correction factor to adjust 'hit percentage' for ship size.

Sources

1. Dive Bomber! by Peter C. Smith, Stackpole Books, (c) 1982 Peter C. Smith
2. Various Internet Forums regarding aircraft
3. Globe security.Org
4. Wikipedia article on V1 Rockets
5. General sources over the years, to numerous to recall.
6. Stuka Pilot by Hans Ulrich Rudel, Blackhouse Publishing (c) 2019
7. Custom Weibull Analysis Program authored by Paul F. Watson

Appendix A: Dive Bomber Saturation Attack Data extracted from Peter C. Smith Dive Bomber!

Appendix B: Weapon Effectiveness:

The purpose of this article was to characterise weapon accuracy, and not the destructive effects of those weapons. But I have chosen to make a few relevant comments. Because dive bombers were accurate, they were typically used during WWII for the following:

• To destroy bridges, rail lines, truck convoys etc.
• To destroy ammunition bunkers
• To destroy hard point defences such as the Maginot line
• To destroy ships at sea or in port

1. “How accurate was the attack expected to be?”
2. “What was the destruction radius of the bomb relative to target?”
3. “What kind of bomb (high explosive or armour piercing) was to be used?
When discussing dive bombers, the size is mostly established. During WWII, naval dive-bombers most commonly carried a single, 500 pound bomb (either high explosive or armour piercing depending on intended target). I welcome information and comments from readers regarding commonly used bombs during land campaigns. Please e-mail such information to me, along with the original source of data so that I might improve this document.

Attacks in open country often cause no real damage if the primary target is missed. Attacks on ships are different. Every square foot of a ship has a purpose — either combat or operationally related. Even operationally related damage often proved fatal (e.g. oil leaking from Bismarck, operational damage to Graf Spee ). For ships, it is largely a matter of ‘hit’ or ‘miss’ although very near misses can cause damage.

WWII Aircraft carriers carried large quantities of bombs, torpedoes and aviation fuel. They were lightly armoured. Three to six bombs usually proved fatal. Merchant ships and destroyers were usually destroyed by a single high explosive bomb. Even heavy cruisers suffered substantial damage from dive bomber attack.

Perhaps the best way to research historical effects of dive-bomber attacks is to review Peter Smith’s book and study various tables that document number of bomb hits and resultant damage. For ships, books such as those by Garzke, Friedman, John Campbell, J.N. Westwood et. al. have detailed descriptions of battle damage for battleships, cruisers, aircraft carriers and destroyers. Information on gunfire damage is often relevant as bombs and shells are simply alternative ways of delivering a weapon.

Appendix C: Authorised Use of This Data & Models:
Free use (without payment) of models and information contained in this article is allowed for game development (whether commercial or not), school or scholarly research PROVIDED this article is referenced and a link to www.dionysus.biz is provided. For printed material, provide the web address in lieu of link. The required format is:

	Information/models used with permission. see www.dionysus.biz  article 
	entitled “Modelling WWII Dive Bombers” Copyright 2020 by Paul F. Watson. 	
	

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