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The full title of this work is "Weaponeering - Conventional Weapon System Effectiveness" by Morris Driels, who teaches at the USN Postgraduate School, and the cover of the edition I have in hand can be seen below.



The book aims to "describe and quantify the methods commonly used to predict the probably of successfully attacking ground targets using air-launched or ground-launched weapons", including "the various methodologies utilized in operational products used widely in the [US military]." Essentially, this boils down to a series of statistical methods to calculate Pk and Ph for various weapons and engagements. 

The author gave the book to my mother, who was a coworker of his at the time, and is of the opinion that Driels is not as smart as he perceives himself to be. But, hey, it's worth a review for friends.

I will unfortunately be quite busy in the next few days, but I have enough spare time tonight to begin a small review of a chapter. I aim to eventually get a full review of the piece done.

Our dear friends @Collimatrix and @N-L-M requested specifically chapter 15 covering mines, and chapter 16 covering target acquisition.

Chapter 15


The mine section covers both land mines and sea mines, and is split roughly in twain along these lines.

The land mine section begins with roughly a page of technical description of AT vs AP, M-Kill vs K-Kill, and lists common US FAmily of SCatterably Mines (FASCAM) systems. The section includes decent representative diagrams. The chapter then proceeds to discuss the specification and planning of minefields, beginning with the mean effective diameter of a mine. Driels discusses a simplified minefield method based on mine density, and then a detailed method.

The simplified method expresses the effectiveness of the minefield as a density value. Diels derives for the release of unitary mines from aircraft

NMines = Fractional coverage in range * fractional coverage in deflection * number of mines released per pass * reliability * number of passes

and for cluster type

NMines = FRange * FDefl * NDispensers * Reliability dispenser * NMines per Dispenser * Reliability Submunition * number of passes

and then exploits the evident geometry to express the Area and Frontal densities. Most useful is the table of suggested minefield densities for Area Denial Artillery Munition and Remote Anti-Armor Mine System, giving the Area and Linear densities required to Disrupt, Turn, Fix, and Block an opponent. 

Whereas the simplistic method expresses effectiveness as a density, the detailed model views the targets and mines individually, assuming the targets are driving directly through the minefield perpendicular to the width and that there is only one casualty and no sympathetic detonations per detonation. The model computes the expected number of targets destroyed by the minefield, beginning with the Mean Effective Diameter and the PEncounter based on distance from the mine. 

Driels derives the number of mines encountered which will be encountered, not avoided, and will engage the target. I can't be arsed to type the equations in full, so here you go.


The section concludes with an example calculation using the detailed mine method. Overall, this shows the strengths and weaknesses of the book fairly well - it is a reasonable derivation of open-source statistical methods for predicting Pk and Ph and the number of sorties required, but US-specific and limited in scope and depth. 

The treatment of Sea Mines  begins by describing the various types and uses of said mines, importantly noting that they have both defensive and offensive uses, and that the presence of the threat of mines is equally important as the actual sinking which occurs. There are three classifications of sea mines, contact, influence, and controlled.

Shallow water mines are treated trivially, considering them equivalent to land mines with Blast Diameter in the place of MED, and assuming that the mines cannot be avoided.

Deep water mines are approached in a similar manner, with the desire to determine the number of mines needed to achieve the required probability of damage, and planning missions from there. Two features of sea mines must be considered, however - mine actuation by passing of the target, and mine damage to the target. The probability of activation is, unfortunately, dependent on the depth of the mine and distance, forming a series of stacked bowls as below.

The mean value of PActivation is the statistical expectation of the curve. Because I don't feel like screencapping another equation, the Width of Seaway where an actuation can occur is qualitatively merely the area under the actuation curve calculated for a specific mine and target combo.

The damage function is also of interest - because we require the mine to both actuate and damage the target, this limits our earlier area under the curve to that area integrated to the limits of the damage function. The selection of mine sensitivity plays a very large role in the effectiveness of our mines. A high setting will lead to many more actuations than damages, which can be indicated by the ratio of the actuation area and the damage area from earlier. Setting the actuation distance equal to the damage distance means that every actuation causes damage, but the probability of actuation is only around 42%. The compromise which selects some Areadamage / Areaactuation of around .8 to .93 is generally preferred. This gives us several useful terms -

PA+D = Reliability * Areadamage / Widthminefield . The probability that the first ship to transit a minefield is referred to as the threat, or

Threat T = 1 - (1 - PA+D)^NMines = 1 - (1 - Reliability * Areadamage / Widthminefield ) which can obviously be solved for NMines to get the desired number of mines for a desired threat level.

Anti-submarine mines are an interesting subset of deep sea mines, as they turn the problem from two-dimensions to three. Driels accounts for this by replacing the mine damage width with the mine damage area, to no one's surprise. Driels claims that the probability of actuation and damage is 

PA/D =  Damage Area / (Width * Depth of minefield). Despite my initial confusion, the reliability term safely reappears in the threat definition below.

T = 1 - (1 - (Reliability * Area damage)/(Width * Depth of minefield))^NMines, with a solution for number of mines for given threat level fairly easily taken out as before.

Lastly, there is a summary of topics for each chapter, though unfortunately they are qualitative descriptions. Including the final derived equations in this part would be a major benefit, but is overlooked. Ah well. They are quite good for review or refreshing the material.

As before, this is a relatively interesting if shallow engagement with the statistical methods to calculate Pk and Ph and the number of sorties required. Going more into detail regarding selecting Threat values or common (unclass) parameters would be interesting, but is lacking. Assuming I don't slack off tomorrow, I should have most or all of the Target Acquisition chapter covered.

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  • 4 weeks later...

I finally got around to doing a review of the Acquisition Chapter. Apologies to @N-L-M for the delay, and apologies in advance for the length.


This topic comes at perhaps an opportune time given a previous conversation with @Sturgeon regarding the Survivability/Lethality Onion.


In this model of lethality, a target must be

First, Seen

Once Seen, Acquired

Once Acquired, Hit

Once Hit, Penetrated

Once Penetrated, Destroyed


With survivability obviously working in reverse. This model has proven quite robust and is extremely popular amongst those studying weapon systems and their survivability. Its terms are generally self explanatory, though it's worth clarifying the first two. Seeing refers to the perception of any given object via some form of sensor. Acquisition, then, refers to processing the perceived object *as a target*, by some form of identification or feature recognition.


That pretentiousness aside, Driels in this chapter will address both topics; the US Joint Munitions Effectiveness Manual (JMEM) considers them in three parts: Target Detection, Target Recognition, and Target Identification under the titular Target Acquisition model. Detection under JMEM’s model correlates most closely with “See” under the Onion, with the remaining two falling under Acquiring the target.


Driels begins by outlining several historical physiological and psychological models for Target Detection, before describing the work by Johnson which is at the core of the US Army’s Acquire model. Terrain, run-in effects, and conversion of range to probability of launch are accounted for, and the factors are combined with the Acquire model to describe the JMEM’s Target Acquisition model.


Unlike the prior chapter, this chapter is much more explicitly focused on Air-to-Surface weaponeering. While the physiological and sensor models will obviously hold true for a variety of detectors, they are used here exclusively to create a model for the range and probability at which an aircraft will detect some ground target. Terrain masking is equally applicable to cases beyond that of Air-to-Surface, but such things as run-in effects and the minimum time taken to bring an aircraft to a required heading are limited in their application.


Published in 1970, the JMEM Air-to-Surface Target Acquisition Manual divides the acquisition process into the aforementioned three steps of detection, recognition, and identification. Their exact definitions have been included here to preserve the granularity.


Blackwell’s research during WW2 began with experiments into the contrast values required to just discriminate circles projected on a screen. Importantly, target size in Blackwell's model is angular in a fashion evident to anyone familiar with MOA.  Beginning with a definition of contrast as

C = abs val of (Luminance of Stimulus - Luminance of Background)/(Luminance of Background)


And of relative contrast as

Cr = actual target contrast / threshold contrast

This definition becomes clear when one examines the situation where Cr = 1, wherein threshold conditions apply and detection probability is 50%. A table of threshold contrast values has been included here, followed by detection probability as a function of the relative contrast.

Practically speaking, to predict detection with this model would require calculating the angular size of the target, then calculating the actual contrast of the target, looking up the threshold contrast, calculating the relative value, and finally determining the probability of detection with the graph. Evidently a complex and lengthy process, these limitations motivated the creation of further detection models.

Overington’s model seeks to correlate the target size and target contrast to the point at which the target is just detectable. This begins with the assumption that the target generates a stimulus between two adjacent retinal receptors between which the boundary of the target and background is located, the magnitude of which will obviously depend on the  magnitude of the contrast. Through a complex series of equations that do not bear reiterating, a relationship is drawn between

.163 * Contrast = K1 * nReceptors + δ

Where K1 is some constant and δ is the minimum stimulus the brain can detect. From this equation, a threshold contrast value can later be obtained. A great deal of care is paid to the amount of receptors which will be stimulated - the minimum even when seeing very small objects (eg stars) is cited as nine.To account for this, a value of

nReceptors = 9.9[(height + width)^2 + .83]^.5 is derived, where height and width of the object are in mrads.


Overington then solves K1 and δ experimentally; they depend on the retinal luminance, which is itself dependent on the scene luminance and the pupil diameter. These equations are not directly solved, and the reader will have to content themselves with the relation of

K1 = 15.4 Retinal Luminance-.5 + .48
δ = .00125 Retinal Luminance -.5 + .0004

With Retinal Luminance equal to pi*pupil diameter squared * scene luminance * 1/4

We can obviously now simplify our earlier equation into

Contrast Threshold = (K1 * nReceptors + δ)/.163

Which is in good agreement with the Blackwell model from earlier. It would later be discovered, however, that these models under-predicted the threshold contrast luminescence. Testing conducted by Johnson in the 1950s wherein observers viewed the side of an M48 (a tank not know for it’s small size, as N-L-M will doubtless attest) showed that the threshold value was higher by around a factor of three compared to the Blackwell and Overington models.


My only brief complaint with this section is that it would benefit from a lengthier comparison between the predicted values and the empirical values for threshold contrast. The history and physiology is interesting, of course, if somewhat dry if all we are given is a simple “it does not work by this factor”.


Johnson’s Frequency-Domain Experiments grew out of these simple “detection” tests, beginning with the fact that mere detection is not sufficient for many military tasks, and that neither a threshold data nor model existed for the tasks of recognition or identification. Initial experiments showed that a nonlinear scaling existed of contrast required with range, which led Johnson to model targets in a frequency rather than spatial domain, best explained visually here.

Each pair of black and white (practically, gray and dark gray) lines is a cycle, and the cycles per milliradian is the cycle frequency. The equivalence between a cycle frequency and a target is constructed as follows.

1. A small image of a military target is projected onto a black screen.
2. An observer is rolled into a position where he can just detect that there is an object on the screen.

3. The image is replaced with a rectangle made of very high cycle frequency bars. The frequency is reduced until the observer can just determine the number of bars.

4. The rectangle is replaced by the image, and the observer is wheeled forward until he just recognizes it. Step 3 is repeated.

5. Step 4 then 3 are repeated with the observer having to identify the target.


The procedure's results have been included here.


This is a strikingly robust and useful model, and has proven sound even when applied to a number of passive sensors such as FLIRs, TVs, and image intensifiers. With it, we can predict acquisition ability of a sensor by measuring its ability to resolve contrast modulated bar patterns. In this passage, Driels discusses an extremely fascinating way of looking at sensors, in a method that’s surprisingly easy to follow given his early work in the chapter. The model seems so simple and robust that one questions why the earlier models are even included, as the Acquire Model to soon be discussed uses Johnson’s work rather than the earlier physiological models.

The US Army’s Acquire Model makes use of Johnson’s Frequency-Domain work, while accounting for significantly more factors. The model begins by calculating the critical dimension (sqrt of the presented area) in mrads, and then selecting an intrinsic contrast value based on the illumination, background, detector, and filters. The attenuation due to atmospheric factors is also taken into account, though the JMEM model only accounts of distance and meteorological factors. Using these factors, the apparent contrast at the sensor is calculated, with

Apparent Contrast = Intrinsic Contrast * Sky to Ground Luminance * e(-3.91 * Range Kilometers / VIS)

VIS represents the atmospheric visibility, and is defined as the range at which contrast is diminished to 2% of Intrinsic Contrast.

The sensor in question is then analyzed using Johnson’s method described earlier. A common measurement standard is a four-bar pattern which can be of varying frequency - ie, the bars can be very few or very many milliradians wide. For a given frequency, the illumination through the sensor is increased from zero until the bars are just able to be distinguished, and this value of contrast is paired with the frequency to construct a Minimum Resolvable Contrast curve. A particular value of frequency for a given apparent contrast on this curve is a Spatial Frequency, yielding

N cycles resolved = SF * critical dimension / Range

Acquire then features a probability for some task (either Detection, Recognition, or Identification) as a function of the ratio between the number of cycles resolved and the number required for a 50% chance of that task being accomplished, included here.


This is a very powerful result, and is again presented quite cleanly and clearly. I appreciate these two passages a great deal more than the earlier parts, especially since they seem most easily applicable to things outside the A2S realm.


Flight profile accounts for the fact that an aircraft does not always approach the target directly down the line of sight, and that several actions must occur for a successful attack even after the target is detected. The aircraft must decide to attack, must roll into and then execute a turn before exiting the turn and operating the weapon system - respectively XD, XRI, XRO, XOP, and RMIN in the diagram here. These will combine with the beginning kinematics and geometry - the turn radius of the aircraft r and nose angle ⍺ - to produce the following RRQ equation.

RRQ = (A cos ⍺ + r sin ⍺) +- Sqrt[ (A cos ⍺ + r sin ⍺ )^2 - (A^2 - B^2)]


This minimum range to maneuver and launch will be included further along in the model in addition to the maximum range for detection. An omission which may be deliberate is the possibility to reverse engineer a target approach to maximize the possibility of detecting the target in time to launch a weapon. 

Searching refers to the process of moving the sensor’s field of view, the solid angle which it can actually “see”, over the entirety of the solid angle the sensor is capable of moving, referred to as the field of regard. Driels places this towards the end of the chapter, but it appears best suited to address earlier. The US Army’s Acquire model expresses the probability of detection as
P = P1 x P2, where P1 is some time-independent probability of detecting the target, and P2 the conditional probability for some amount of time, best explained below.


Terrain has the effect of blocking almost all sensors used by aircraft, with the particular quandary that terrain can vary quite rapidly and unpredictably. (Something anyone attempting to learn land navigation can attest to.) Driels constructs a workable model for the angle at which an aircraft’s sensors are unmasked as follows - for a given “type” of terrain, place an observer at some random point. The observer measures the angle to the highest terrain feature along a given bearing, which is the unmask angle. This repeats this for the entire circle, producing a cumulative probability distribution of the unmask angles, and the process may be repeated for a variety of terrain types to any desired level of granularity.

The JMEM target acquisition model covers flat farmland, smooth desert, rolling farmland w/ close forests, rolling desert, flat farmland with close forests, gently rolling hills, rough desert, and sharply rolling hills with trees, though it should be evident that any particular terrain type could be easily calculated. The omission of any form of urban terrain is puzzling, however, and the question of which existing terrain to model it with is thought provoking. Perhaps sharply rolling hills with trees? That this 2004 book does not cover the acquisition of targets in urban terrain is no great discredit, as it has likely been accounted for in more recent versions of the JMEM acquisition model, but it certainly merits further discussion moving forward to a doubtless more urbanized battlefield.


There are now models in hand for the two major limits on range of detection, terrain and visibility, and from this Driels proceeds to construct a conversion between range and the probability of detection and launch.


This equation, and the cumulative probability that one can derive from it, accounts for not only the distance the aircraft must close to before detecting the target, but also the time taken to search the volume available to the sensor suite and the minimum time/distance required to maneuver the aircraft to launch position. The constant K accounts for the skill level of the pilot, Pmax is assumed to be 1, R is the smaller of the unmask range or the visibility detection range established earlier, RRQ is the minimum range to maneuver and drop. A series of calculations have been performed in the chart here - these seem to be far lower than occurs in reality, potentially due to the choice in parameters.  


Driels then details the usage of the JMEM Target Acquisition Model, a screen of which is included here. (Note that PL is significantly higher than his table earlier.) Inputs to the model can be taken from the Joint Air-to-Surface Weaponeering System (JAWS), as well as additional information regarding the target, vision conditions, weapon trajectory, and launcher kinematics - the latter two obviously determining RRQ.


In Summary, the chapter examines physiological models of detection by Blackwell and Johnson, addresses their implementation in the US Army’s Acquire model, and then details the Joint Munitions Effectiveness Manual’s use of Acquire and the additional information its model includes. This chapter is interesting and offers a great deal of unexploited potential - the models are all extremely fascinating, and I can easily imagine their direct applicability towards S2S or passive A2A detection. Crucially, however, the acquisition models all appear to completely negate *active* sensors, possibly for reasons of confidentiality. Still, the fundamentals behind the two-way radar equation aren’t that complex, and could easily be slotted into the existing maximum range of visibility parameter. Beyond that, it is an interesting chapter, and one of the most insightful in the book.

I think I'm done with the book for now. I may do some simulation of infantry fires by plagiarizing Driels' direct fire chapter, but that is a tale for another day.

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