Models, Gaming, and Simulation Session 3

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Models, Gaming, and Simulation - Session 3

• Physical Models of Attrition

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Topics

• The role of attrition in combat modeling
• Physical models of attrition
• Accuracy Models
• Lethality Models
• Multi-Shot Engagement Models
• Next Session Attrition for Low-Resolution Models

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The Role of Attrition in Combat Models
well-modeled

• The history of rigorous analysis of combat dates
  from Lanchesters (1914) models of attrition
• Analytic stand-alone models of combat have
  focused almost exclusively on the attrition
  processes
• There is a wide range of quality in modeling of
  the various combat processes

• tank-on-tank attrition
• other direct- and indirect-fire attrition
• direct-fire target acquisition
• communications
• terrain
• air / air-defense
• sensor acquisition processes
• command and control
• intelligence fusion
• human decision-making
• planning
• effects of training
• motivation, esprit, courage, etc

poorly-modeled
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The Role of Attrition in Combat Models

• We tend to model best what we can quantify (for
  example, attrition)
• WARNING Dont ascribe undue importance to a
  factor just because you can model it well (e.g.,
  human factors like motivation and decision-making
  are probably just as important in battle as good
  weapons).
• Consider Effects-based vice Attrition-based
  Models.

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The Role of Attrition in Combat Models

• Engineering Models
• GOAL Determine how accurate and what damage a
  given shot does to a given target, i.e., tabulate
  bias and dispersion, PHS and PKH, for all
  shooter/target pairings.
• High-Resolution Models
• GOAL Determine what damage a given firing weapon
  does to a given target.
• Medium-Resolution Models
• GOAL Determine how many vehicles by type in a
  given unit are damaged by a given firing unit.
• Low-Resolution Models
• GOAL Determine how a given target units
  effectiveness measure is degraded by
  participation in a battle.

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The Role of Attrition in Combat Models

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Physical Models of Attrition

• Engineering Models
• "Flyout" models
• Represent ballistics, weather, and other
  technical factors explicitly over time-of-flight
  to compute PH in a specific situation
• AMSAA has responsibility in the Army for
  providing this data JMEMS provides similar data
  for all Services.
• Damage models represent PKH in great detail.
• "Cell Models" predicts what effects a specific
  munition will have on a specific vehicle.
• Developed by Survivability and Lethality Analysis
  Division (SLAD) of Army Research Lab (ARL)
  (formerly Ballistics Research Lab or BRL)
• High-Resolution Models
• Use AMSAA or JMEMS data as input (i.e., PHS and
  PKH )
• Assess attrition shot by shot.
• Single-shot accuracy
• Single-shot lethality
• Multiple-shot assessment

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Single-Shot Accuracy

• PROBLEM Find point of impact (if any) of round
  on its target.
• ASSUMPTION The projectile impact point is a
  random variable with a normal probability
  distribution (empirically shown to be a good
  assumption).

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Single-Shot Accuracy Measures

• Normal Parameters. For 2D target
• "Side View" (i.e., direct-fire weapon)
• Elevation error
• Deflection error
• "Top View" (i.e., indirect-fire weapon)
• Range error
• Deflection error
• DEFINE
• Bias ?x , ?y
• Dispersion ?x , ?y
• Circular Error Probable (CEP) - radial distance
  from aim point within which half of rounds will
  land (assume ?x ?y)
• Probable Error - distance in deflection (for x)
  within which half of rounds will land (similar
  for y)

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Single-Shot Accuracy 1-Dimensional Target

NOTE Z is available in tabular form in any
Statistics text see Normal Distribution.
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Single-Shot Accuracy 1-Dimensional
Small Target

• ASSUME Small Target (i.e., L ltlt ?), and ? 0
• THEREFORE PDF is constant locally, so PH is
  approximated by a rectangle with width 2L and
  height f(0).

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Single-Shot Accuracy
2-Dimensional Target

• ASSUME Round impact is randomly distributed with
  bivariate normal pdf

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Single-Shot Accuracy Circular Target

• ASSUME Circular Target (Radius R)
• ASSUME Circular Impact Distribution (?x????y)
• ASSUME No Bias (?x????y 0)
• CONCLUDE

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Single-Shot Accuracy Aligned Rectilinear Target

• ASSUME Aligned Rectangular Target
• (GT line is parallel to X-axis)
• ASSUME ? 0 (deflection and range errors are
  independent)
• ASSUME X N(?x , ?x), Y N(?y , ?y)
• THEN
• PH Prob-LX lt X lt LX Prob-LY lt Y lt LY
• Now compute each factor as in 1-dimensional case.

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Single-Shot Accuracy - Polya-Williams
Approximation

• ASSUME Aligned Rectangular Target
• (with dimensions 2Lx by 2Ly)
• ASSUME X N(?x , ?x), Y N(?y , ?y)
• ASSUME ? 0 (deflection and range errors are
  independent)
• ASSUME ?x ?y 0 (no bias)
• APPROXIMATE PH BY

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Single-Shot Accuracy - 2D Small-Target

• ASSUME Small Target (i.e., Lx ltlt ?x, Ly ltlt ?y),
  ? 0, and ? 0
• f(x)
• F(x)
  dxdy
• PH
  dxdy

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Single-Shot Accuracy - Monte Carlo Evaluation
Using Bias Dispersion

• Consult AMSAA or JMEMS tables for ?x, ?y , ?x ,
  ?y for a given munition and situation.
• Generate two random numbers X N(?x , ?x), Y
  N(?y , ?y)
• Compare (X,Y) with target geometry to determine
  if round hit target.
• If goal is to assess whether a given shot hit,
  you are done.
• If goal is to compute PH (e.g., to build a table
  of PH's), repeat steps 2 and 3 to get a
  sufficiently large sample PH is the fraction of
  rounds that hit.

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Single-Shot Accuracy - Monte Carlo Evaluation
Using Precomputed PH

• Consult AMSAA or JMEMS tables for PH for given
  munition and firer-target situation.
• Generate random number X U(0, 1).
• If X lt PH, then assess a hit otherwise assess a
  miss.

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Direct-Fire Accuracy Example

• An infantry fighting vehicle (IFV) has the
  following frontal profile
• A hit in area 1 will
• produce a firepower kill
• A hit in area 2 will
• produce a catastrophic kill
• A hit in area 3 will
• produce a mobility kill
• A hit in other areas will
• produce no permanent effect
• Assess the IFVs vulnerability when engaged with
  a frontal shot whose impact point is modeled as a
  random variable pair (X,Y) BVN(0,0,.5,.5,0).
• Using the below list of pseudo random numbers as
  needed, simulate the first round to determine
  which type of kill, if any, occurs. (.8554,
  .2287, .6659, .8243, .6840, .0430, .8598, .2381,
  .5035, .2723)

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Direct-Fire Accuracy Example

• 1) Do a Monte Carlo simulation of impact
• point with origin centered on the target,
• then compare impact point with target
• profile to calculate where it hit.
• 2) Determine X coordinate of impact point
• Enter the Normal Table with .8554
• Find Z-1 1.06
• Note that Z-1 ((x-?x)/?x
• Solve for x in 1.06 (x-0)/.5
• x .53
• Determine the Y coordinate of the impact
  point (using RN .2287)
• Normal Table goes from .5000 to .9999, but
  Normal Dist. is symmetric,
• so compute 1.0-.2287.7713, and change sign of
  resulting Y coordinate.
• Interpolating between .75 and .74, gives Z-1
  .743.
• Solve for y in -.743(y-0)/.5 gives y-.3715
• 3) Round hits area 4, so no kill is assessed.

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Single-Shot Lethality

• Impact Projectiles
• (direct fire and smart munitions)
• PK PH PKH
• (where PKH is tabulated from a SLAD Cell
  Model)
• Fragmenting Projectiles
• (area fire)
• PK f(miss distance)

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Single-Shot Lethality - Kill Categories

• K-Kill Catastrophic Kill
• M-Kill Mobility Kill
• F-Kill Firepower Kill
• MF-Kill Mob Firepower Kill, usually gt K-Kill
• P-Kill Personnel kill (crew and passengers)
• No-Kill No damage due to hit.

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Single-Shot Lethality - Area Weapons

• ASSUME Impact coordinates (X,Y) BVN(?x, ?y ,
  ?x, ?y , 0)
• ASSUME Target located at (0,0)
• ASSUME Radial symmetry of damage
• Focus on miss distance r define damage as a
  function of .
  Define damage function D(r) as the
  probability that a target is killed by a weapon
  when miss distance is r.
• General form
• or in polar coordinates,

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Single-Shot Lethality - Area Weapons Lethal Area

• ASSUME Target location is uniformly distributed
  over a large area A.
• THEN
• and
• Since A is large,
• Introducing polar coordinates
• Define "Lethal Area" aL as
• Then

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Single-Shot Lethality -Area Weapons Lethal Radius

• ASSUME D(r) is non-increasing
• DEFINE RL as a random variable such that any
  target within RL is killed.
• Since D(r) is the probability that a target will
  be killed if round impacts at distance r, we can
  say
• D(r) ProbrltR
• So PDF of RL is
• fR(r) -dD(r)/dr
• It can be shown that
• aL ? ERL2

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Single-Shot Lethality -Area Weapons
PK

• Example Damage Function
• Example Lethal Area
• aL .8(25?) .4(75?)
• 50?
• What if we wanted an equivalent Damage Function
  such that either PK 1 or PK 0
• PK 1
• PK (?r2) 50?
• r2 50
• rL 5 2

.8
y
.4
x
5m
10m
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Single-Shot Lethality - Area Weapons Cookie
Cutter Weapon

• Let RL be constant then Damage Function is
• D(r)
• and aL ?RL2
• ASSUME Circular BVN Impact Distribution, No
  Bias, i.e., (X,Y)
  BVN(??????????, 0)
• THEN
• ALTERNATELY
• where CEP ???? 2 ln 2 1.1774 ?

P0 for r lt RL 0 for r gt RL
???
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Single-Shot Lethality - Area Weapons Cookie
Cutter Weapon

• ALTERNATE VIEW Think of trying to hit a circular
  target of radius RL. If a hit occurs, then there
  is a P0 probability of a kill. Then
• PK P0 PH
• where PH is computed using offset circular normal
  tables.

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Single-Shot Lethality - Area Weapons Carleton
Weapon

• ASSUME exponential damage function with constant
  scale factor b, i.e.,
• THEN aL 2 ? b2
• ASSUME Impact coordinates (X,Y) BVN(?x, ?y ,
  ?x, ?y, 0) and Target located at (0,0)
• THEN
• ASSUME Circular impact distribution, no bias,
  then

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Multi-Round Engagement Models

• Independent Shots (usually direct fire)
• n shots fired by n firers at same intended aim
  point
• Salvo
• n shots by 1 firer at same aim point
• Adjust-Fire ("Shoot-adjust-shoot", or
  "Burst-on-target")
• n shots by 1 firer, modifying aim point
• Pattern Volley
• n shots by n firers at n distinct aimpoints
  offset from target location
• Shoot-Look-Shoot
• 2 or more shots by 1 firer at same aim point,
  noting target effect after each shot

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Multi-Round Engagement Models - Independent Shots

• ASSUME n I.I.D shots
• OR, ASSUME n independent (but differently
  distributed) shots
• i.e., recompute PK for each subsequent shot

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Multi-Round Engagement Models - Adjust-Fire

• Model must describe method of adjusting
• Example Burst-on-target model might specify
  that
• ?x,n -Xn-1 / 4
• ?y,n -Yn-1 / 2
• Two approaches
• 1. Simulate step-by-step, i.e., compute each
  successive PK if kill is realized, stop
  otherwise continue.
• 2. Approximate with independence assumption

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Multi-Round Engagement Models - Pattern Volley,
Single Target

• Step-wise simulation is most practical approach
• Determine Target Location Error (TLE)
• Simulation each round
• Determine aim error
• Let Bias TLE aim error
• Determine dispersion
• Look up PH in offset circular normal tables
• Compute PK using Cell Model (Point Fire) or
  Damage Function (Area Fire)
• Generate a random variable and compare with PK to
  determine if a kill occurs.
• Assess damage in one of two ways
• If any round killed target, assess a kill
• Accumulate damage round to round

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Multi-Round Engagement Models - Pattern Volley,
Multiple Targets

• Target Coverage Model ("Superquickie" algorithm)
• Determine impact area in which rounds impact
• Determine number of targets of a given type which
  lie inside impact area
• Assumptions uniform target distribution and
  uniform round impact
• Target Damage Model
• Integrate damage function over various random
  factors, e.g., target distribution, aim error
  distribution, etc.
• Very complex, but may be useful for a very
  high-resolution, ad hoc model for a specific
  purpose.

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Multi-Round Engagement Models - Pattern Volley,
Multiple Targets Target Coverage Model

• Let
• N number of targets
• NK number targets killed
• AI impact area
• aL lethal area
• at area of each individual target
• F(n) fraction of AI covered by aL of at least 1
  of n rounds
• Assume
• 1. AI gtgt aL
• 2. at ltlt aL
• 3. Targets distributed uniformly throughout AI
• 4. Rounds impact uniformly and independently
  throughout AI

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Multi-Round Engagement Models - Pattern Volley,
Multiple Targets Target Coverage Model

• Then
• aL / AI fraction of AI covered by one round
• 1 - aL / AI fraction of AI not covered by a
  given round
• F(n) 1 - (1 - aL / AI )n (does not require
  assumption 1)
• F(n) (requires
  assumption 1)
• Note that F(n) is the fraction of target elements
  killed, so
• ENK

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Multi-Round Engagement Models - Pattern Volley,
Multiple Targets Target Coverage Model

• Remember that lethal area is
• (intuitively, this is the total amount of
  lethality under the damage function curve)
• Approximate the fragmentation pattern of each
  round by a cookie-cutter weapon whose total
  lethality is aL.
• For exponential damage function,
• aL 2 ? b2

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Multi-Round Engagement Models - Pattern Volley,
Multiple Targets Target Coverage Model Example

• Six-round volley is fired with linear sheaf as
  shown
• Each round has Carleton (exponential) damage
  function with scale factor b5.0
• Target is infantry company with 200 soldiers
  deployed in a 200m by 100m formation (assume
  uniform distribution of soldiers in formation)
• ATGT 20000
• AVOLLEY 5000
• AI 5000m2
• aL 2?b2 50? 157.0795m2
• N 200(5000/20000) 50tgts
• F(n) 1 - (1 157.0795/5000)6
• ENK N F(n)
• 8.715 kills

100m
Target
50m
200m
100m
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Whats Next?

• Modeling attrition in low-resolution simulations
  (two or three sessions).
• Then, modeling target acquisition.
• Then, modeling C2 and other battlefield functions
  (i.e., how to model units of combat systems
  interoperating on the battlefield).

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Generating a Normal Random Number - Two Methods

• Method 1
• Probability Integral Transform If Xfx(x) with
  CDF Fx(X) and Fx(X) is continuous, then Fx(X) U
  is uniformly distributed over the interval (0,1).
  Conversely, if UU(0,1), then XFx-1has CDF
  Fx(X).
• Practical Importance you can generate normal
  random numbers using a uniform random number
  generator (e.g., random(i) in C) and Normal (Z)
  tables.
• Approach Use the Z table backwards
• generate a value u of UU(0,1)
• find u in the body of the Z table
• read the corresponding value of X
• Method 2
• Generate a bivariate normal random pair using a
  mathematical formula
• X (-2 ln(U1(0,1))1/2 (sin(2pU2(0,1))
• Y (-2 ln(U1(0,1))1/2 (cos(2pU2(0,1))
• This is a useful coding approach