Attachment 4 ARSP-2 VOLI EDB V1 E (2470).pdf

NATO UNCLASSIFIED

NATO STANDARD

ARSP–2 Volume I

GUIDANCE ON THE DEVELOPMENT OF

WEAPON DANGER AREAS/ZONES

PROBABILISTIC METHODOLOGY – GENERAL

PRINCIPLES

Edition B Version 1

NOVEMBER 2015

NORTH ATLANTIC TREATY ORGANIZATION

ALLIED RANGE SAFETY PUBLICATION

Published by the

NATO STANDARDIZATION OFFICE (NSO)

© NATO/OTAN

INTENTIONALLY BLANK

NORTH ATLANTIC TREATY ORGANIZATION (NATO)

NATO STANDARDIZATION ORGANIZATION (NSO)

NATO LETTER OF PROMULGATION

19 November 2015

1. The enclosed Allied Range Safety Publication ARSP-2 Volume I, Edition B, Version 1, GUIDANCE ON THE DEVELOPMENT OF WEAPON DANGER

AREAS/ZONES FOR USE BY NATO FORCES PROBABILISTIC

METHODOLOGY - GENERAL PRINICIPLES, which has been approved by the nations in the MCLSB, is promulgated herewith. The agreement of interested nations to use this publication is recorded in STANAG 2470.

2. ARSP-2 VOL I, Edition B, Version 1, is effective upon receipt and supersedes ARSP-2 VOL I, Edition A, Version 1, which shall be destroyed in accordance with the local procedure for the destruction of documents.

3. No part of this publication may be reproduced, stored in a retrieval system, used commercially, adapted, or transmitted in any form or by any means, electronic, mechanical, photo-copying, recording or otherwise, without the prior perrnission of the publisher. With the exception of commercial sales, this does not apply to member or partner nations, or NATO commands and bodies.

This publication shall be handled in accordance with C-M(2002)60.

Edvardas MAZEIKIS Major General, L TUAF Director, NATO Standardization Office

ARSP-2 VOL I

I Edition B Version 1

RESERVED FOR NATIONAL LETTER OF PROMULGATION

II Edition B Version 1

III Edition B Version 1

RECORD OF RESERVATIONS

CHAPTER RECORD OF RESERVATIONS BY NATIONS

Note: The reservations listed on this page include only those that were recorded at time of promulgation and may not be complete. Refer to the NATO Standardization Document Database for the complete list of existing reservations.

IV Edition B Version 1

V Edition B Version 1

RECORD OF SPECIFIC RESERVATIONS

[nation] [detail of reservation]

DNK Implementation will be reconsidered in the event that the present use of deterministic methods becomes stochastic.

POL 1. Poland reserves the right not to apply the provisions included in the STANAG, resulting from the current possibilities of armament systems, especially with regard to the use of compatible computer software and the related aspects of training security.

2. The postponing results from the need to develop a national response with regard to the STANAGs referenced in ARSP-2 Ed. B:

3606/ARSP-1, Vol. II, 2921, 2402.

Note: The reservations listed on this page include only those that were recorded at time of promulgation and may not be complete. Refer to the NATO Standardization Document Database for the complete list of existing reservations.

VI Edition B Version 1

VII Edition B Version 1

TABLE OF CONTENTS

Chapter 1 General

0101 Introduction 1-1 0102 Aim 1-1 0103 Scope 1-1 0104 Vocabulary and abbreviations 1-1 0105 Related documents 1-2

Chapter 2 Risk

0201 Introduction 2-1 0202 Probability, frequency, and units 2-1

0203 Some common measures of risk for WDA/Z 2-2

Chapter 3 Illustrations and examples of the probabilistic methodology

0301 Introduction 3-1

0302 Illustration 1 — an unguided weapon launched with varying elevation 3-1

0303 Illustration 2 — an unguided weapon launched with varying elevation 3-1

0304 Illustration 3 — an unguided weapon with a fragmenting warhead 3-2

0305 Illustration 4 — an unguided weapon with ricochet 3-3

0306 Further examples 3-3

0307 Guided weapons 3-5

0308 Composite WDA/Z 3-5

Chapter 4 General principles of the probabilistic methodology

0401 Introduction 4-1 0402 Deterministic, probabilistic, and hybrid methodologies 4-1 0403 Sensitivity, variability, and uncertainty 4-1 0404 The distribution of final resting places — “probability of escape” 4-2 0405 Individual or collective risk — frequency of death or injury 4-2 0406 Weapon Danger Zones — three and four dimensional problems 4-3 0407 Composite WDA/Z 4-4

Chapter 5 Probability/frequency calculations

0501 Introduction 5-1 0502 Probabilistic models 5-1 0503 Monte Carlo simulation 5-1 0504 Importance sampling 5-2

0505 Data cubes 5-3 0506 Combined calculations 5-4 0507 Calculation of probability of hit, injury, or death 5-4

0508 Histograms and frequency polygons 5-5

0509 Combining histograms or frequency polygons 5-6

0510 Smearing histograms or frequency polygons 5-7

VIII Edition B Version 1

Chapter 6 Calculation of hazard and risk from probability distributions

0601 Introduction 6-1 0602 Calculations for a simple polygon 6-1

0603 Calculations for a simple polyhedron 6-2

Chapter 7 Development of WDA/Z from probability distributions

0701 Introduction 7-1 0702 Contours 7-1 0703 Convex hulls and specified outlines 7-1

0704 Contours vs. convex hulls and specified outlines 7-2

0705 Composite WDA 7-1 0706 Population dependent WDA 7-2

0707 Two dimensional subsets of three dimensional results 7-2 0708 WDZ or ADH for zero risk 7-2

Annex A Probability calculations — analytic methods

A01 Introduction A-1 A02 Transformation of variables — example 1 A-1 A03 Transformation of variables — example 2 A-2 A04 Liouville’s equation and a Fokker-Planck equation for a point-mass A-3

Annex B Application of transformation of variables to a simple point-mass

B01 Introduction B-1 B02 Trajectory equations B-1

B03 Trajectory solution B-1

B04 Transformation of variables B-2 B05 Variability — probability contours B-2

Annex C Application of simulation to a simple point-mass problem

C01 Introduction C-1 C02 Variability C-1

C03 Uncertainty C-1

Annex D Options for calculation of hazard and risk from density functions

D01 Introduction D-1 D02 Hazard and risk using impact density D-2

D03 Hazard and risk using probability of hit/injury/death D-3

Annex E Options for development of WDA from density functions

E01 Introduction E-1 E02 WDA using impact density E-2

E03 WDA using probability of escape E-2

E04 WDA using frequency of escape E-4

E05 WDA using event frequency of escape E-5

E06 WDA using annual frequency of escape E-6

IX Edition B Version 1

E07 WDA using probability of hit/injury/death E-7

E08 WDA using frequency of hit/injury/death E-9

E09 WDA using event individual risk of hit/injury/death E-10

E10 WDA using event collective risk of hit/injury/death E-11

E11 WDA using annual individual risk of hit/injury/death E-13

E12 WDA using annual collective risk of hit/injury/death E-14

Lexicon L-1

References R-1

LIST OF FIGURES

Chapter 1 General

Figure 1.1 — Framework of Allied Range Safety Publications 1-3

Chapter 2 Risk

Figure 2.1 — Risk management framework. 2-5

Chapter 3 Illustrations and examples of the probabilistic methodology

Figure 3.1 — Trajectories and impact ranges corresponding to two elevations for Example 1.

3-7

Figure 3.2 — Trajectories showing impact locations corresponding to different elevations and azimuths for Example 2.

3-7

Figure 3.3 — Combined WDA and BSD with a single BSD outside for Example 3. 3-8

Figure 3.4 — Trajectories showing impact locations, and subsequent ricochets, corresponding to different elevations and azimuths for Example 4.

3-8

Figure 3.5 — WDAs for a single firing position (FP) and two target positions (TP1 and TP2) with an exposed population (E).

3-9

Chapter 4 General principles of the probabilistic methodology

Chapter 5 Probability/frequency calculations

Figure 5.1 — Domains of definition for P(A) and P(B). 5-8

Figure 5.2 — Domain of definition for P(A and B) over domains of definition for P(A) and P(B).

5-8

Figure 5.3 — Domain of definition for P(A or B), P(A xor B), min[P(A),P(B)] and max[P(A),P(B)] over domains of definition for P(A) and P(B).

5-9

Chapter 6 Calculation of hazard and risk from probability distributions

Figure 6.1 — A simple polygon over the grid defining the frequency polygon. 6-3

Figure 6.2 — Partition of a single edge of the polygon into line segments. 6-3

Figure 6.3 — Quadrilateral Q and triangle T used in integration over grid cells. 6-4

Chapter 7 Development of WDA/Z from probability distributions

Figure 7.1 — A convex hull corresponding to a contour in two dimensions. 7-3

Figure 7.2 — A specified shape covering a contour in two dimensions. 7-3

X Edition B Version 1

Figure 7.3 — A specified shape partially covering a contour in two dimensions. 7-4

Annex A Probability calculations — analytic methods

Annex B Application of transformation of variables to a simple point-mass problem

Annex C Application of simulation to a simple point-mass problem

Figure C.1 — Variability problem — comparison of fractiles obtained by weighted simulation with analytic solutions.

C-4

Figure C.2 — Uncertainty problem — comparison of fractiles obtained by weighted simulation using sample with reference solutions.

C-5

Figure C.3 — Uncertainty problem — the first 25 of a sample of M = 1000 simulations 1 in 1 000 000 contours compared with the population and sample 1 in 1 000 000 contours.

C-6

Figure C.4 — Uncertainty problem — comparison of confidence limits obtained by weighted simulation with the population and sample1 in 1 000 000 contours.

C-7

1-1 Edition B Version 1

CHAPTER 1

GENERAL

0101. INTRODUCTION

1. A Weapon Danger Area (WDA) is that area associated with firing a weapon where the risk of death or injury exceeds some threshold. The risk outside the WDA does not exceed this threshold and hence the risk to people outside the WDA is acceptable or tolerable. A Weapon Danger Zone (WDZ) extends this into three dimensions. Traditionally, WDA have been developed using deterministic methodology and WDA are extended into WDZ by using a constant height above the WDA. In both cases the level of risk associated with the area or zone has been assessed as acceptable or tolerable, but has not been explicitly quantified. In order to quantify the levels of risk we have to use a probabilistic methodology.

2. Weapon Danger Area/Zones (WDA/Z) encompass the ground and airspace for lateral and vertical containment of projectiles, fragments, debris, and components resulting from the firing, launching, and/or detonation of ordnance. WDA/Z account for weapon accuracy, failures, ricochets, and broaches/porpoising of a specific weapon/munition type. WDA/Z developed using deterministic principles, where a number of worst case assumptions are used, are usually bigger than they need to be, and this often results in the use of large areas and can constrain training. The probabilistic methodology can be used to handle more complex situations:

a. Specific range danger areas/zones (RDA/Z) to account, for example, for local terrain and local met conditions can be developed;

a. The methodology can applied to the specific situation using realistic data so that range space can be optimized;

b. Ranges can be designed to contain projectiles;

c. Probabilistic analyses can provide information that can be used for other risk management purposes, for example to quantify the risk to range infrastructure;

d. Probabilistic analyses can provide information that can be used to diagnose problems for conceptual or existing ranges.

3. It is important to note that the use of a probabilistic methodology is not a universal remedy.

Whilst is has many advantages over deterministic methodology, probabilistic models have to be developed and data needs to be gathered and analyzed for use in these models and this may not be a simple process.

0102. AIM

1. Whereas WDA are traditionally classified by weapon type and role, the general principles described here apply to all weapon systems and no such distinctions need to be introduced. This aim of this document is to describe these general principles so that they may be applied for all weapon systems and appropriate WDA/Z may be developed.

0103. SCOPE

1. This publication is relevant to the development of WDA/Z for all weapon systems. Although the description here is based on ballistic weapons the principles apply to all weapon systems and lasers (either used as part of a system or as weapons). Specific information for various categories of weapon systems is given in related publications.

0104. VOCABULARY AND ABBREVIATIONS

2. A list of terms and abbreviations used in this publication are provided in the Lexicon.

1-2 Edition B Version 1

0105. RELATED DOCUMENTS

1. This is one of a sequence of Allied Range Safety Publications (ARSPs) that are concerned with the development of WDA/Z for a variety of weapon systems for use by NATO forces in a variety of roles. The framework is shown in Figure 1.1. Brief descriptions of each ARSP are given below:

a. Volumes in STANAG 2401 (Reference 1) with ARSP-1 cover the deterministic methodology:

b. Volume I (Reference 2) contains a description of the factors that are relevant to the use of unguided weapons.

(1) Volume II (Reference 3) contains a description of the application of the factors from Volume I, and provides generic danger area outlines together with nation dependent numerical values for the factors.

b. Volumes in STANAG 2470 (Reference 4) with ARSP-2 cover the probabilistic methodology:

(1) Volume II (Reference 5) contains a description of the application of these principles to unguided weapons. It includes descriptions, and in some cases detailed specifications, of the models that may be used when applying the probabilistic methodology to the factors in ARSP-1 Volume I.

(2) Volume III (Reference 6) contains a description of the application of these principles to guided weapons (GW).

(3) Volume IV contains a description of the application of these principles to unmanned aerial vehicles (UAVs). This is an update of STANAG 2402, Edition 2 (Reference 7).

c. Volumes in ARSP-3 cover the acquisition and analysis of data for use with both deterministic and probabilistic methodologies:

(1) Volume I contains a description of trials procedures and data analysis for aimer deviations.

(2) Volume II contains a description of trials procedures and data analysis for free-flight data.

(3) Volume III contains a description of trials procedures and data analysis for fragmentation data.

(4) Volume IV contains a description of trials procedures and data analysis for impact and post-impact models.

d. STANAG 3606 (Reference 8) with ARSP-4 (Reference 9) covers the factors relevant to lasers and the application of deterministic and probabilistic methods to lasers.

1-3 Edition B Version 1

Figure 1.1 — Framework of Allied Range Safety Publications.

1-4 Edition B Version 1

2-1 Edition B Version 1

CHAPTER 2

RISK

0201. INTRODUCTION

1. The determination of a WDA/Z and its use as an exclusion area/zone is part of a risk management process that aims to ensure that the risk to the general public, civilian, participating and/or non-participating military personnel arising from the use of weapon systems is kept to an acceptable or tolerable level.

2. A framework for risk management (References 10 and 11) is shown in Figure 2.1. This shows the place of risk analysis, which consists of hazard identification and risk estimation, within the overall framework. In this publication we are only directly concerned with the two risk analysis stages.

3. Concerns over the levels of risk that are considered acceptable or tolerable, which are relevant in the risk evaluation stage, are dealt with as national issues and are prescribed in national documents.

4. In the hazard identification stage the aim is to identify all the hazards associated with the use of a particular weapon system. In many cases the hazards are generic to particular classes of weapon system and those relevant are identified in standard lists. These generic lists should be developed, documented, and maintained. Where particular weapon systems do not conform to a general class and/or present additional hazards a specific document may be produced for each individual weapon system.

5. An example of a generic list is ARSP-1 Volume I (Reference 1), which provides a list of factors (i.e. hazards) that should be considered when developing WDA/Z for unguided weapon systems. This was produced, and is maintained, as a separate document from the document that provides the WDA/Z. For other generic weapon systems the list may be included in the same document. Guided weapon systems fit into the second category as each guided weapon system tends to have unique guidance and aerodynamic behaviour. Irrespective of the location of the list it is important to realize that the hazard list is the same for both deterministic and probabilistic methodologies.

6. It is in the risk estimation stage that the difference between deterministic and probabilistic methodologies is seen. In the probabilistic methodology a direct estimate of the risk is attempted.

With the deterministic methodology worst case assumptions may be used to bound the risk without actually estimating it directly. In addition heuristic methods that appear to provide an appropriate level of risk are used.

0202. PROBABILITY, FREQUENCY, AND UNITS

1. Before describing some common measures of risk some of the terminology of risk is described.

There are many different terminologies used for risk and it is often difficult to get agreement on the terminology that should be used in a particular situation. Two examples of the definition of risk are:

a. Risk is the probability of a particular adverse consequence (Reference 12);

b. Risk is a combination of frequency and adverse consequence (Reference 13).

2. Problems arise when one definition is provided and an alternative is used in calculations.

Similar confusion arises between the terms hazard and risk. It is important to use an appropriate definition and ensure that what is calculated corresponds to the definition. Much of this confusion can be eliminated by (1) understanding the difference between probability and frequency, and (2) always using units when quoting probabilities or frequencies, i.e. not quoting numbers such as 1 in 1 000 000 on their own.

3. Probability is defined as “a real number in the scale 0 to 1 attached to a random event” (Reference 14). It can be related to a long-term relative frequency of occurrence or to a degree of belief that an event will occur — both approaches lead to much the same principles — but the relative

2-2 Edition B Version 1 frequency interpretation is the most common. Frequencies occur as a result of the calculation of expectations, which combine probabilities per initiating event and the number of initiating events. A key difference between probability and expectation is that probabilities are constrained to the scale 0 to 1 whilst frequencies are not, and hence any quantity that can be greater than 1 is a frequency.

4. A probability (or frequency) is a probability (or frequency) of something and there are some units involved, e.g. 1 death in 1 000 000 person years. All the various components of a calculation have their own units and both the numbers and units are combined appropriately. Often the units are ignored, when their use would indicate that something is wrong with the calculation. Sometimes the result is a risk but does not match the definition used, and sometimes the result is not a risk at all.

5. As an example of the use, and combination, of units consider the calculation for a probability of death per round fired. This could arise as a result of the use of a weapon system firing a single projectile and the calculation could combine two terms as:

deaths deaths hits round hit round

D R D H H RP P P

(2.1) where D HP may depend on the part of the body hit and the presence, or otherwise, of body armour;

and H RP is calculated assuming people are present the whole time the weapon system is in use.

0203. SOME COMMON MEASURES OF RISK FOR WDA/Z

1. Here we describe some measures of risk that are in common use for developing WDA/Z. The simplest is probability of escape. The most general is frequency of death from which, a range of similar measures can be derived.

2. A list of measures together with options for calculating hazard and risk (from Chapter 6) is provided in Annex D and with options for the development of WDA from density functions (from Chapter 7) is provided in Annex E.

3. Probability, or frequency, of escapes from the WDA/Z in escapes per firing.

a. Traditionally this has been used with weapon systems for developing WDA/Z. When we assume that the weapon system fires a single projectile that remains in one piece and a single projectile does not kill more than one person the result is a probability. Where multiple projectiles result from a single firing, e.g. for projectiles that break up on impact with the terrain, or projectiles that fragment, the result is a frequency as a single firing can result in multiple escapes.

b. This is a hazard (i.e. not a risk) as the escape from the WDA/Z is not necessarily an adverse consequence. However, it can be converted to a risk by adding terms, which are then assumed to have value 1. For example, if we add two terms for hits per escape and deaths per hit we obtain:

deaths deaths hits escapes round hit escape round

D R D H H E E RP P P P

. (2.2)

c. When worst case values of 1 are taken for the additional terms it is seen that this places a bound on the risk of death in terms of deaths per round:

2-3 Edition B Version 1

/ /1 1 deaths deaths hits escapes round hit escape round

D R E RP P

. (2.3)

d. Probability of escape and any measures of risk derived from it are properties of the WDA/Z. The levels of hazard or risk apply to the exterior region between the WDA/Z boundary and the boundary of the zero energy area/zone. They are only indirectly functions of position — generally the risk decreases from the boundary of the WDA/Z to zero at the boundary of the zero energy area/zone but the calculation of probability of escape cannot demonstrate this.

4. Frequency of death in deaths per person year, known as annual individual risk of death (IR) (Reference 15).

a. This is the most common measure of risk of death used in compiling national statistics, where it is calculated on an actuarial basis as the ratio of deaths from a particular activity in a year divided by the number of people that participated in that activity in that year.

b. It is the accepted measure of risk within GBR for accidental death whilst at work and is the preferred measure used by GBR Government Departments for setting safety standards (References 16 and 17) and is being adopted by some other nations for other defence related purposes (Reference 18).

c. The calculation for the frequency of deaths per person year is derived from (2.1) by adding two terms:

deaths rounds deaths hits years person year year hit round person year

D PY R Y D H H R Y PYF F P P E

(2.4) where the additional term R YF is the frequency, i.e. number, of rounds fired in a year, and Y P YE is the exposure i.e. the proportion of time that people are present when the weapon system is in use.

d. Although it is a frequency, as the calculation can produce numbers greater than 1, it is often referred to as a probability because any activity that produced numbers greater than 1 would obviously be unacceptable.

5. Individual risk of death (IR) can also be used on a per event basis.

e. The calculation for the frequency of deaths per event is derived from (2.4) by considering events instead of years and removing the exposure term:

D PE R E D H H RF F P P E

deaths rounds deaths hits 1 person event event hit round person (2.5)

6. Individual risk is a function of position and it varies widely over the WDA/Z — it is high directly in front of the firer and generally decreases to zero at the boundary of the zero energy area/zone, and the calculation of individual risk will determine this. In most cases IR is zero before the boundary of the zero energy area/zone because projectiles near the boundary generally have insufficient energy to cause death.

7. One advantage of the use of individual risk over the use of probability of escape is that the examination of the different terms in the calculation provides a direct link with risk mitigation measures.

If the overall risk is too high, any combination of reduction in each of the four terms, in (2.4), can be

2-4 Edition B Version 1 investigated. This already happens within most range management systems but the link with the risk estimation process is not evident:

a. The use of the WDA/Z as a restricted zone, keeping people out of the area or zone, reduces the exposure term Y P YE to zero and the risk inside the WDA/Z is zero.

b. The use of body armour reduces the probability of death or injury given a hit term D HP .

c. The probability of hit term H RP can be reduced by range design, by for example the addition of baffles or barriers, which stop projectiles entering specific locations within the original WDA/Z.

d. Finally, the frequency of firing R YF can simply be reduced, or limited, to a level that produces an appropriate overall level of risk.

8. Where IR is concerned with the risk to an individual, collective risk (CR) considers all people exposed to the hazard. It is measured in terms of the expected number of deaths, either per event or per year. For example, the annual collective risk of death is given by:

D Y D PY

P

F F deaths deaths person year person year

(2.6)

9. It is also easy to derive other measures of risk from risk of death. We may decide to use frequency of injury, for example in injuries per person year, or frequency of death, for example in hits per event.

a. The first of these is derived from (2.4) by replacing deaths per hit by injuries per hit:

injuries rounds injuries hits years person year year hit round person year

D PY R Y D H H R Y PYF F P P E

(2.7)

b. The second is derived from (2.5) by using the value of 1 for deaths per hit. This produces an upper bound on the individual risk of death and is considered a cautious approach that avoids the requirement to specify wounding models:

D PE R E H R Y PEF F P E

/ / / /1 deaths rounds deaths hits 1 person event event hit round person . (2.8)

2-5 Edition B Version 1

Figure 2.1 — Risk management framework (showing the place of risk analysis, which consists of hazard identification and risk estimation, within the overall framework)

2-6 Edition B Version 1

3-1 Edition B Version 1

CHAPTER 3

ILLUSTRATIONS AND EXAMPLES OF THE PROBABILISTIC METHODOLOGY

0301. INTRODUCTION

1. A purely mathematical presentation of the principles of the probabilistic methodology is neither easy to understand nor necessary. The principles are illustrated here using examples involving unrealistically simple weapon systems before some of the factors that have to be considered in more complicated situations are described.

2. We start by considering only the calculation of WDAs for a single static firing position. A problem of unguided weapon free flight where the only random input is the elevation is described in clause 302. We show how a WDA can be developed for this problem before complicating factors are added one at a time and the effect on the development of a WDA is examined in clauses 303 to 305.

Clause 306 contains some simple examples that involve complications, such as moving platforms and moving targets. The determination of WDA/Zs for guided weapons (GW) is covered in clause 307.

Finally the determination of composite WDA/Zs, which may arise from the use of multiple firing positions and/or different weapon systems, is described in clause 308.

3. Whilst reading this chapter it may be useful to consider how WDA/Zs would be obtained using a deterministic methodology. In some cases the deterministic methodology is difficult to apply, in others the use of a purely probabilistic methodology is not possible and we end up with a hybrid methodology.

1. The simplest case that can be considered is that of an unguided weapon where all parameters are constant apart from the elevation, which is randomly distributed about some mean value. The terrain is flat, the azimuth is zero, there is no wind, the trajectory is planar (i.e. there is no drift), and hence the impact location for a given elevation is given by the range from the firing position. The maximum elevation is below that needed for maximum range, each unique elevation corresponds to a unique impact position, and range increases as elevation increases.

2. The WDA is taken to be from the firing point to the range that is exceeded by only 1 in 1 000 000 rounds and the WDA then contains 999 999 in 1 000 000 rounds. Once we know the distribution of elevations we have to determine this range in order to develop the WDA.

3. In Figure 3.1 two trajectories 1T and 2T with ranges 1 2R R corresponding to two elevations

1 2E E are shown. As trajectories do not cross each other it is evident that all elevations between

1E and 2E produce trajectories between 1T and 2T with ranges between 1R and 2R . If the probability that an elevation is greater than 2E is 2P , this is also the probability that a trajectory is above 2T and a range is greater than 2R . The WDA is found by finding the elevation that is exceeded in 1 in 1 000 000 rounds and then computing the range at this elevation.

1. A simple modification, a randomly varying azimuth, is now made to illustration 1 that introduces a significant complication. This azimuth is randomly distributed about a mean value of zero. For a unique elevation and azimuth the impact location is again unique but is now specified by two coordinates:

cos sin x R A z R A

(3.1)

3-2 Edition B Version 1 where A is the azimuth and the range R depends on elevation as before.

2. In Figure 3.2 a number of trajectories with impact locations corresponding to different elevations and azimuths are shown. As before the trajectories do not cross and the relationship between ranges and elevations is the same i.e. all elevations between 1E and 2E produce ranges between 1R and

2R .The relationship between azimuth and cross-range position z is similar — for a given range, all elevations between 1A and 2A produce cross-range positions between 1 1cosz R A and

2 2cosz R A .

3. In trying to develop a WDA we now have a more complicated problem. If we specify a probability of escape, such as 1 in 1 000 000 firings, how are we to choose the area that corresponds to this? This question occurs because there is now no unique area that corresponds to a given probability. There are a number of ways to overcome this:

a. Treat range and azimuth independently and choose a probability for each such that the combination is 1 in 1 000 000.

b. This is not unique as we could choose to use maximum range (thus covering all ranges) and azimuth limits to contain 999 999 in 1 000 000 firings. This is still not unique as the azimuth limits are not determined and a further apparently arbitrary choice has to be made.

c. The usual way of choosing is to use a limit on range that covers 0.999 999 and limits on azimuth that cover the same proportion i.e. 0.999 999 . The combined area then covers the required proportion. As with the previous choice this is still not unique as the azimuth limits are not determined.

d. The two common methods of choosing the azimuth limits are to (a) use lower and upper limits such that the probability of being outside either is the same, and (b) choose lower and upper limits such that the probability density has the same value at both limits.

Method (a) corresponds to the classical method for choosing central confidence intervals (Reference 19), whilst (b) corresponds to that used in Bayesian methods (Reference 20).

Note that for a symmetric probability density (a) and (b) are equivalent.

e. Use a contour in the probability function and join it back to the launch point. This is equivalent to a two dimensional application as in a. (3) above. It has the property that the area chosen is a minimum.

1. We complicate illustration 1 by adding a simple fragmenting warhead that is assumed to function on impact and has a constant circular fragmentation pattern.

2. The usual method for dealing with fragmenting warheads is to add a burst safety distance (BSD) to the existing WDA. It is often assumed that the original WDA and the new combined WDA and BSD have the same frequency associated with them. That this is not the case as can be seen by examination of the following example.

3. Assume that we have chosen the original WDA to cover all but 1 in 1 000 000 firings and use a BSD that encloses all fragments from the warhead event. The resulting combined WDA and BSD is shown in Figure 3.3 together with a single BSD for another warhead event. Even though this additional warhead event occurs with probability below 1 in 1 000 000 when the warhead produces a large number of fragments it is now possible to have the frequency of escape being higher than the original 1 in 1 000 000 firings.

3-3 Edition B Version 1

4. Only the probabilistic methodology can be used to develop a correct WDA for this case. The distribution of the final resting places of fragments has to be constructed and the WDA chosen to contain all but 1 fragment in 1 000 000 firings of the weapon.

0305. ILLUSTRATION 4 — AN UNGUIDED WEAPON WITH RICOCHET

1. We now complicate illustration 2 by adding ricochet with the terrain on impact. Now ricochets may occur, depending on the impact conditions. Example trajectories including first ricochets after impact are illustrated in Figure 3.4.

2. As the ricochet process is random there is no apparent pattern to the locations of the final resting places. Where a complication arises in illustration 1 when low and high angle fire are present together because the launch conditions cannot be determined from the impact conditions, here the situation is much more complicated. There is no ordered relationship between the initial elevation and the final down-range position or between the initial azimuth and final cross-range position. It would appear that any final resting place can be reached in an infinite number of ways from any combination of the initial angles.

3. In the deterministic methodology, opening angles to allow for ricochet are applied from the boundaries of the region containing the initial impact area. As with the use of a combined WDA and BSD there is no guarantee that the probability or frequency associated with the resulting WDA has any particular relationship to that of the original WDA.

4. As with illustration 3, only the probabilistic methodology can be used to develop a WDA for this case. The distribution of the final resting places of projectile has to be constructed and the WDA chosen to contain all but 1 projectile in 1 000 000 firings of the weapon.

0306. FURTHER EXAMPLES

1. The four illustrations above have shown how the probabilistic methodology is adapted to handle simple complications. Even with these simple problems it becomes clear that the application of the probabilistic method is not straightforward. In this clause we briefly describe some other examples with additional complications that illustrate a number of things — the probabilistic data that is required to use the methodology — the differences between the deterministic and probabilistic methodologies

— and problems where the use of a purely probabilistic methodology is difficult and the use of a deterministic method is easier.

2. The first example involves the use of explosives in training, which can involve training for operational Explosive Ordnance Disposal (EOD) or training in use of demolition charges, where we wish to determine the WDA/Z. Some of the additional factors that have to be considered are:

a. The ammunition or explosive is deliberately initiated.

b. For EOD this will almost certainly involve non-design mode functioning of the ammunition on, or in, the ground. How does the ammunition function? We usually only have data for design mode functioning in free air.

c. For EOD the position and orientation of the ammunition may be unusual. The burst safety distances that are used in other circumstances are determined for the orientations and velocities that arise in standard firing scenarios and here we have to use the data in situations where it may not be valid.

d. The outcome may depend on other weapons/mitigations that are involved. For demolition devices the break-up of the structure may produce fragments that travel further than those from the device.

3. The next example involves the use of a laser, say a range finder, or target designator, where we wish to determine the laser danger area/zone. The development of a danger area for a laser is similar to that for a ballistic weapon, except that it involves eye damage rather than death or injury caused by being hit by projectiles. Some of the factors considered are:

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a. The laser parameters — analogous to the aerodynamic parameters of a projectile.

b. The pointing error for the laser system — analogous to the aimer deviations for a ballistic weapon system.

c. The probability of a person being irradiated — analogous to being hit by a projectile.

4. Some of the additional factors that have to be considered are:

a. The probability of an irradiated person looking at the laser.

b. The probability that atmospheric effects, such as scintillation, increase the radiant exposure entering the eye.

c. The probability of the received exposure causing ocular damage.

5. The next example involves a fixed wing aircraft flying over a range, with nominal altitude, run-in heading, dive angle, and airspeed. The weapon system is fired at a fixed point target, and we wish to determine the WDA/Z either over time or as a function of release envelope. Some of the additional factors that have to be considered are:

a. The altitude, run-in heading, dive angle, and airspeed are all subject to error and their distributions have to be specified.

b. How is the weapon system aimed? Is it manual or is a targeting system used? This will determine whether the aimer error depends on the pilot or is a function of the fire-control system.

c. Once we can develop WDA/Z for known firing situations, how do we apply these? Do we want a single composite WDA/Z that covers all scenarios that could occur, or a sequence of WDA/Z that change with time and firing situation?

6. The next example involves a naval platform engaged in gun fire support, firing from ship to shore, and we wish to determine the WDA/Z for a target area. Some of the additional factors that have to be considered are:

a. The target is an area and not just a point. How do we deal with the distribution of aiming points within this target area? Do we specify a distribution for these?

b. The weapon system will involve a number of electronic systems on board the platform.

How does the interaction of these contribute to the aimer error? A model for aimer error may be difficult to derive.

c. The movement of the platform is determined by the sea state, which is beyond the control of the platform and this will contribute to the aimer error. Do we have to determine the WDA/Z as a function of sea state?

7. The next example involves a helicopter, hovering or manoeuvring over a range, firing a machine gun out of the side door at a set of fixed targets, and we wish to determine the WDA/Z. Some of the additional factors that have to be considered are:

a. The weapon system is unstable. What additional allowance do we make for aimer error?

b. The projectiles experience wind shear as they emerge from the boundary layer surrounding the helicopter. What are the actual meteorological conditions and what effect do these have?

c. What effect does the downwash from the rotor have? With some projectiles a proportion of those fired is thrown of course by the downwash. How do we allow for this?

8. The next example involves the determination of a “firing box”. A weapon system is being fired at a known point target and the safety boundary is known — it could for instance be the boundary of a

3-5 Edition B Version 1 range. We wish to determine those firing positions that we can use without the risk being inappropriate at the safety boundary.

a. This requires the development of a WDA/Z for all valid firing positions and keeping a list of those where the WDA/Z results in the risk being appropriate.

b. The checking of the infinite number of firing positions is not practical and one approach is to use a grid over the area of interest. With an assumption that the relationship between the risk levels from the resulting WDA/Z are continuous this marks out the firing box but it should be noted that the “box” may not be a regular shape or indeed a single area or zone.

9. A similar example is that of determining a “target box”. A weapon system is being fired from a known firing position and the safety boundary is known. We wish to determine the target positions that we can use without the risk of projectiles escaping outside the safety boundary. The same procedure used in the previous example can be adopted with the result being again a “box” that may not be a regular shape or indeed a single area or zone.

0307. GUIDED WEAPONS

1. The distribution of the trajectories and subsequent final resting places for unguided weapons are subject to disturbances that are principally continuous. For guided weapons on the other hand many of the disturbances to which they are subject are discrete. These arise because of individual faults, for example in the guidance system, that change the nature of the trajectories.

2. In order to analyse the behaviour and determine the trajectories and any associated probabilities from them, it is necessary to handle the outcome of each combination of discrete faults.

The distributions for each outcome are then combined with frequencies corresponding to their probability of occurrence obtained from the underlying fault tree analysis.

3. This situation is beyond the deterministic methodology and hence a probabilistic approach is used. The probabilistic methodology used for guided weapons is identical to that described in this publication e.g. the use of simulation to obtain an approximation to the probability distribution of the final resting places.

0308. COMPOSITE WDA/Z

1. The examples listed above are all for a single firing position and a single target position. The term composite is used to refer to situations with multiple firing positions and/or multiple target positions and is used to indicate that the WDA/Zs are made up of a number of separate parts.

2. With the deterministic methodology a composite WDA/Z is usually developed by merely overlaying the individual WDA/Z for each firing and/or target position and this same approach can be used with the probabilistic methodology. However, it is important to realize that the risk for the composite WDA/Z is not the same as that for the individual WDA/Zs, though of course it may still be at an acceptable or tolerable level.

3. As an example, consider the use of a single firing position with two target positions illustrated in Figure 3.5. If the risks for the exposed population from firing at target position 1 and target position 2 are 1R E and 2R E individually, what is the risk from firing at the two target positions together?

The answer depends on the measure of risk (individual risk or probability of escape) being used and exact meaning of the word together (at the same instantaneous time, or mutually exclusively).

4. Where individual risk is used and 1 1R F and 2 2R F are estimated from the use of 1N and

2N rounds per year and the two targets are used at the same instantaneous time the individual risk is

12 1 2 1 2 1 2, ,F E F E F E S N N t F E F E . (3.2)

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5. The last term, which represents the risk from being at risk by firing at target position 1 and target position 2 at the same time is the product of the two frequencies scaled to account for the fact that 1F and 2F are annual frequencies.

6. The scale factor 1 2, ,S N N t depends on the numbers of rounds fired and the time interval we consider — note that without the scale factor the product represents the frequency with which someone is hit by rounds fired at target 1 and target 2 in the same year.

7. At locations near the target positions this term could be significant, but at most locations outside the individual WDA/Z where the frequencies are small this term is negligible.

a. Where individual risk is used and 1 1R F and 2 2R F are estimated from the use of N rounds per year and the two targets are used mutually exclusively with these rounds split between the two target positions, i.e. 1 2N N N the individual risk is

1 2 1 2 1 2max , N N F E F E F E F E F E

N N . (3.3)

b. Where probability of escape is used it does not matter whether the two targets are used at the same instantaneous time or mutually exclusively. The probability of escape is the sum of the expected number of escapes from firing at each target divided by the number of rounds fired. If the number of rounds fired at targets 1 and 2 in the ratio 1 2:N N the probability of escape is

N P N P

N N

1 1 2 2

1 2

(3.4)

8. It is evident that the development of composite WDA/Z using overlays of individual WDA/Z is not necessarily a simple process even for the case of a single firing position and two target positions. For more complex cases, such as that of where sets of firing positions at different ranges are in use with a single set of target positions the calculation of risk using algebra is not feasible. However, with the probabilistic methodology the probabilities or frequencies can be combined using a computer program and WDA/Z can be developed for these cases in exactly the same way that they are developed for the simple cases.

9. The probabilistic methodology can also handle problems where different weapon systems, for example small arms and medium calibre systems, are used at the same time. A more interesting example is that for a weapon system where a laser is used with a ballistic system. With the deterministic methodology individual WDA/Z are used for the laser and the ballistic system are simply overlaid whereas the probabilistic methodology could be used to develop a composite WDA/Z for the laser and ballistic system considered together. This would require some common criterion for risk to be used as the combination of the use of probability of escape for a laser pulse and individual risk of death for the ballistic system would not make sense.

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Figure 3.1 — Trajectories and impact ranges corresponding to two elevations for Example 1.

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Figure 3.2 — Trajectories showing impact locations corresponding to different elevations and azimuths for Example 2.

E2

E1

T2

T1

R1 R2

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Figure 3.3 — Combined WDA and BSD with a single BSD outside for Example 3

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Figure 3.4 — Trajectories showing impact locations, and subsequent ricochets, corresponding to different elevations and azimuths for Example 4

WDA + BSD BSD

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Figure 3.5 — WDAs for a single firing position (FP) and two target positions (TP1 and TP2) with an exposed population (E).

FP

TP1

E

TP2

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CHAPTER 4

GENERAL PRINCIPLES OF THE PROBABILISTIC METHODOLOGY

0401. INTRODUCTION

1. After the illustrations given in Chapter 3 a more formal presentation of the principles of the probabilistic methodology is given here. We start by describing briefly the deterministic methodology before describing the probabilistic methodology. For practical applications the methodology used is rarely a pure implementation of the probabilistic method, because the development of probabilistic models and the acquisition and analysis of data for all events is often not possible, or indeed feasible.

An important observation of the use of a hybrid methodology where some deterministic components and some probabilistic ones are used together is given.

0402. DETERMINISTIC, PROBABILISTIC, AND HYBRID METHODOLOGIES

1. With the deterministic methodology a sequence of “worst case” components is used to model the complete problem. It is important to recognize that these are worst in the sense that they yield the most severe solution to the complete problem and may not necessarily be the worst for the individual component. One obvious example of this is the launch elevation for an unguided projectile — if the elevations include that for maximum range use of the minimum and maximum angles will not reproduce the maximum range.

2. With the probabilistic methodology all random components are described in terms of probability distributions and these are used to construct a probability or frequency distribution for the complete problem. Practical methods for calculating this distribution are described in Chapter 5, whilst some theoretical methods are described in Annex A.

3. In hybrid models a mix of the two methodologies is used. Some components are treated probabilistically and some are handled deterministically. As with the pure deterministic methodology worst cases are used for the deterministic components — in the sense that they yield the most severe probabilistic solution to the complete problem.

0403. SENSITIVITY, VARIABILITY, AND UNCERTAINTY

1. When using any mathematical model that contains components where the parameters that describe the model of the component are not certain it is important to understand the sensitivity of the model to the various inputs. A sensitivity analysis estimates the rate of change of the output to changes in each of the inputs.

2. Variability — formally called aleatory uncertainty — arises because variables are random and individual values vary about the expected value. Variability may be characterized by the variance of the distribution of a variable and can be illustrated by calculating fractiles of the distribution.

3. Uncertainty — formally called epistemic uncertainty — arises because we have incomplete knowledge, either because our models are not adequate or because we do not know the true distributions or parameter values.

4. In the unlikely event that there are no…