SYSTEMS AND METHODS FOR ESTIMATING INTERCEPT CAPABILITIES OF ACTIVE MANEUVERABLE WEAPONS CROSS-REFERENCE TO RELATED APPLICATION [01] This application claims the benefit of United States Provisional Patent Application No. 63/004,316 filed April 2, 2020, the entire contents of which is incorporated herein by reference. FIELD [02] This application relates generally to assessing the intercept capabilities of aerial or nautical weapons, such as missiles or torpedoes, and in particular to assessing the effect and effectiveness of countermeasures in diverting such weapons to protect assets, such as ships. It has a more general application in any instances where there is concern for the possibility of proximity between two objects in motion relative to each other, including but not limited to physical collision. This might include any combination of aircraft, ground vehicles and surface ships, and any cause of proximity, whether intentional or not. BACKGROUND [03] The fundamental purpose of missile countermeasures is to induce a heading error in the missile’s flight trajectory in order to prevent an intercept of a potential target. Consider a sea- skimming radar homing missile engaged by a non-specific angle deception countermeasure able to induce a heading error. If the heading error is sufficiently large, then the missile is unable to execute a maneuver to correct the flight trajectory and intercept the target. However, because conventional countermeasures systems have not provided an effective and accurate indication of when an in-flight missile has been sufficiently diverted in order to guarantee that the missile cannot intercept the target, determining the real-time, in-situ protection offered by a missile countermeasures system is a problem. Without a practical measure of real-time effectiveness, in a conventional countermeasures system it is generally difficult, if not impossible, to optimize the selection and/or control of the deployment of countermeasures to maximize the probability of survival for any or all ships threatened by an in-flight missile. [04] In view of the above, it would be beneficial to be able to assess the effect and effectiveness of missile countermeasures, including active onboard jamming and the use of passive and active offboard decoys. SUMMARY [05] According to a first broad aspect, the present disclosure provides a method for assessing ability of a maneuverable first object to intercept an area or volume containing a second object. For example, in some embodiments a method according to the first broad aspect may include obtaining or estimating guidance properties of the first object relative to the area or volume containing the second object, determining an intercept corridor defining a subset of points in three-dimensional space from which is it is physically possible for the first object to intercept the area or volume containing the second object, and producing an indication of the intercept corridor. In some cases, the guidance properties include at least speed (scalar) and/or velocity (vector) data for the first object and a maximum lateral acceleration limit for the first object. [06] According to a second broad aspect, the present disclosure provides another method for assessing ability of a maneuverable first object to intercept an area or volume containing a second object. For example, in some embodiments a method according to the second broad aspect may include obtaining or estimating guidance properties of the first object relative to the area or volume containing the second object, determining whether the first object is capable of intercepting the area or volume containing the second object based on the guidance properties, and producing an indication of a result of the determination whether the first object is capable of intercepting the area or volume containing the second object. In some cases, the guidance properties include at least: an observed rate of bearing change of the first object relative to the area or volume containing the second object; speed and/or velocity of the first object; and a maximum lateral acceleration limit for the first object. In some cases, the guidance properties may also or instead include at least: an observed heading error of the first object relative to the area or volume containing the second object; speed and/or velocity of the first object; and the maximum lateral acceleration limit for the first object. [07] According to a third broad aspect, the present disclosure provides an apparatus for assessing ability of a maneuverable first object to intercept an area or volume containing a second object. For example, the apparatus may include an engagement monitoring and/or management system comprising at least one processor configured with instructions that when executed implement a method according to the first and/or second broad aspects described above or other methods as described herein. An example might be a system for managing the position and velocity of an offboard decoy intended to prevent the missile from intercepting one or more potential targets. [08] In some embodiments of one or more of the foregoing aspects, the first object may be a guided weapon, such as a missile or a torpedo. [09] In some embodiments of one or more of the foregoing aspects, the second object may be a fixed object or a maneuverable object, such as an aircraft or a ship. [10] Other aspects and features of embodiments of the present disclosure will become apparent to those ordinarily skilled in the art upon review of the following description. BRIEF DESCRIPTION OF THE DRAWINGS [11] The foregoing summary, as well as the following detailed description of illustrative embodiments of the present application, will be better understood when read in conjunction with the appended drawings. For the purposes of illustrating the present application, there is shown in the drawings illustrative embodiments of the disclosure. It should be understood, however, that the application is not limited to the precise arrangements and instrumentalities shown. In the drawings: [12] Figs. 1A and 1B are isometric and plan views, respectively, of a geometry for derivation of minimum heading error for guaranteed missile miss for an engagement scenario with a point target; [13] Fig. 2 illustrates a plan view of a geometry for derivation of minimum heading error for guaranteed missile miss for an engagement scenario with a distributed target; [14] Figs.3A and 3B are plan views illustrating an effect of a rotating coordinate frame as a missile approaches a ship/sensor; [15] Fig.4 is a plot of required bearing rate for missile soft kill vs. range for several protection field widths; [16] Fig. 5 is a plot of normalized bearing rate correction term vs. range for several protection field widths; [17] Figs. 6A and 6B are plan and isometric views, respectively, that provide qualitative visualizations of a missile intercept corridor for an engagement scenario with zero heading error; [18] Figs. 6C-6E are isometric, ship and top down views, respectively, of a missile intercept corridor for an engagement scenario with a missile guided in both azimuth and elevation; [19] Fig. 7 is a plan view that provides a qualitative visualization of a missile intercept corridor for an engagement scenario with nonzero heading error; [20] Figs. 8A and 8B are plan views that provide a qualitative visualization of an effect of miss distance on soft kill range; [21] Fig. 9 is a plan view of a geometry for derivation of anticipatory jammer turn-off (AJTO) conditions; [22] Fig. 10 is a plan view of an example of the concept of an intercept corridor applied to multiple ships; [23] Fig. 11 illustrates an example of an alternative definition of soft kill defined by excessive flight time; and [24] Fig. 12 illustrates an example of an apparatus for assessing ability of a maneuverable first object to intercept an area or volume containing a second object, according to an embodiment of the present disclosure. DETAILED DESCRIPTION [25] The following detailed description refers to the accompanying drawings. The same reference numbers in different drawings may identify the same or similar elements. [26] It is theoretically possible to define conditions under which a guided missile is physically unable to intercept its intended target, based on limitations of its agility. A mathematical formalization of these conditions is potentially applicable to assessing the effect and effectiveness of missile countermeasures, including active onboard jamming and the use of passive and active offboard decoys. The fundamental purpose of missile countermeasures is to induce a heading error in the missile’s flight trajectory, to prevent an intercept. The concepts and mathematical approach are based on the limitations of the missile’s agility, or achievable turn rate. Because of these limitations some heading errors are, in principle, unrecoverable for the missile to successfully intercept the target. [27] Consider a sea-skimming radar homing missile engaged by a non-specific angle deception countermeasure able to induce a heading error. If the heading error is sufficiently large, then the missile is unable to execute a maneuver to correct the flight trajectory and intercept the target. Under these conditions and ignoring the possibility that the missile may come about and re-engage the target after a miss has occurred, the missile is guaranteed to miss the target. It is possible to introduce the concept of guaranteed missile miss conditions, and to base this concept solely on performance limitations of the missile airframe. Point Target Case - Missile Range Formulation [28] As noted above, the failure of a missile to intercept its intended target may be considered to be guaranteed if the countermeasure induces a heading error incrementally greater than that which would require the missile to execute a maximum acceleration turn to intercept the target (ignoring the possibility that the missile may come about and re-engage the target after a miss has occurred). A geometry for describing range and heading error conditions for a guaranteed missile miss is presented in figs.1A and 1B, which are isometric and plan views, respectively, of an engagement scenario with a point target. An expression shall be derived using this geometry for the range r′ at which a given heading error results in a guaranteed missile miss. [29] Referring to figs. 1A and 1B, the range r′ between the missile and the target is given by the following expression, for a missile in a maximum lateral acceleration turn: where r′ = radial range between missile and target, r = minimum missile turn radius, and θ = off-axis look angle to missile from target (figs.1A-1B) [30] The maximum heading error for which the missile is just able to intercept the target is designated φ. The calculation of φ assumes that perfect guidance information is available at the instant of interest and that the missile is instantly able to achieve maximum turn rate toward the target. This assumption means that the value of φ is conservative, i.e. represents a worst-case scenario in favour of the missile. For the purposes of this analysis and considering a target with zero width, φ also refers to the minimum heading error for which the missile misses the target, here referred to as a marginal miss. The minimum required heading error φ for a guaranteed missile miss in terms of off-axis look angle θ is given by the following expression: [31] The case of φ ≥ π ⁄ 2 corresponds to a heading error so large that the missile is receding from the target. In the context of this investigation, this condition is heuristically defined to be a state of successful soft kill. For this reason, the condition φ ≥ π ⁄ 2 is disallowed in the following mathematical treatment. [32] Isolating θ in eqn (2) and substituting into eqn (1) yields the following expressions: [33] The missile’s minimum turn radius r is given by the following expression: where: v _m = missile velocity a _max = missile maximum lateral acceleration. [34] Substituting eqn (4) into eqn (3) yields the following expression for the range at which a given heading error results in a missile miss: [35] Eqn (5) defines the radial range r′ between the missile and the target at which a given heading error φ results in a marginal miss for a point target. Derivation of a Bearing Rate Requirement for Missile Miss [36] Eqn. (5) can be used to derive a relationship between conditions of a guaranteed missile miss and the rate of change of missile bearing as observed from the point target. If an inbound missile develops heading error, there is a cross-range component to its velocity vector as seen by the point target. This causes the apparent bearing of the missile to change over time as seen by the target. An expression shall now be derived for the observed bearing rate of the missile under the conditions of a marginal miss. [37] Consider the case where θ ‹ π ⁄ 2 (ref. eqn (2)). Eqn (5) can be re-cast to express the heading error required to cause a marginal miss in terms of the current missile radial range from the target. Isolating φ from eqn (5) yields the following expression: [38] Following the geometry of fig. 1, the instantaneous rate of change of missile bearing as seen by the point target is given by: where v _t = component of missile velocity tangential to the point target = _{missile bearing rate (rad/sec)} [39] Substituting eqn (6) into eqn (7) yields: eqn (8) [40] The sine of the sum of two angles can be rewritten using the following identity: [41] Applying eqn (9) to eqn (8) yields: [42] Eqn (10) can be simplified to the following expression: [43] According to eqn (11) and considering the case of a point target, the missile bearing rate which must be observed for a guaranteed missile miss is independent of missile range. It implies that missile defeat relative to a point target can be determined by a single observable variable, if the missile velocity v _m and agility a _max are known. Both of these parameters are generally known or estimable a priori to the engagement. Additionally, missile defeat is heuristically defined to have occurred if the heading error becomes so large that the missile begins receding from the target (ref. eqn (2), fig. 1). In some cases, missile velocity may be obtained via an onboard shipboard radar that provides a ground truth measurement of missile velocity. In other cases the missile velocity (vector) and/or speed (scalar) may be estimated or assumed by some other means. For example, missile speeds tend to cluster around a few values which are determined by tradeoffs of fuel economy, weight penalty, financial cost per round, lethality, etc. One such common value is high subsonic (around Mach 0.95 [~300 m/sec]), another is a supersonic (around Mach 2.5 [~800 m/sec]). Other speed bins/groupings are also possible. In general, the equations, and their equivalent formulations in alternative co-ordinate systems, allow a determination of the missile instantaneous soft kill state for whatever missile velocity is used, irrespective of whether it is measured or estimated. [44] It is useful to briefly consider a numerical example: considering a missile velocity of 300 m/sec and a maximum lateral acceleration of 2.5 gravities (24.53 m/sec ² ), the required bearing rate is 2.3 degrees/sec. If the countermeasure can cause the rate of change of missile bearing to exceed 2.3 degrees/sec at any time during the engagement, the missile cannot intercept the point target. This bearing rate is judged to be readily measured or readily extractable using existing bearing measurement technology. Implications for Weaving Missiles [45] It is possible that some missiles may execute weaving maneuvers during their terminal homing phase as a countermeasure to hard kill weapons systems. Eqn (11) indirectly implies that such missiles are potentially at a disadvantage relative to a non-weaving missile. In the context of this analysis and the concept of a guaranteed missile miss, the effect of intentional weaving maneuvers is indistinguishable from flight deviations caused by angle deception countermeasures. Weave maneuvers implicitly induce a cross-range component of missile velocity relative to the target, and cause the missile to approach the missile intercept corridor boundary. The trajectory deviations caused by the ECM can be thought of as additional to this motion. If the flight deviations caused by the ECM are linearly superimposed on the flight deviations comprising the weave maneuvers, and if they are both instantaneously in the same direction, then the observed bearing rate will be greater than that observed if the weave maneuvers were absent and the missile will more closely approach the missile intercept corridor boundary. During intervals where the weave maneuver is in the same direction as the flight disturbance caused by the ECM, the ECM is required to cause smaller flight deviations in order to cross the missile intercept corridor boundary than if the missile were in steady-flight. Nonzero Protection Field Width [46] Eqn (7) relates the rate of change of missile bearing observed at the target to the heading error φ in order that the missile miss a point target. Eqn (7) can be modified to reflect a different and potentially more useful definition of a missile intercept: to be judged an intercept, the missile must pass through a finite width gate containing the target. Alternatively, in order for a missile flight to be judged a miss, the missile may not pass through the finite width gate, here referred to as the protection field. The concept of a protection field corresponds to a minimum acceptable miss distance for a missile flight to be judged a miss. It is interesting to note that the missile intercept corridor can also be used to define conditions under which an intercept is unavoidable, i.e. conditions under which the missile must pass through the protection field. [47] In this section an equation shall be derived for the missile bearing rate which must be observed at a centrally located sensor in order that the missile flight trajectory is a marginal miss with respect to the boundaries of the protection field. The geometry for the derivation of minimum heading error for guaranteed missile miss for a distributed target is presented in fig.2. [48] Note that this is not a fixed coordinate frame, it is a coordinate frame which rotates during the engagement such that the missile is always on the z-axis (i.e. y = 0), as shown in figs. 3A and 3B, which illustrate the effect of a rotating coordinate frame as a missile approaches a ship/sensor. This means that under certain extreme circumstances (e.g. a near miss by the missile) the missile flight trajectory could impinge on a circle of radius d centered on the ship, in a fixed coordinate frame. This is a known minor deficiency of the derivation which follows. [49] The heading error φ in eqn (6) can be rewritten following the geometry of fig. 2. The cosine law has the following form: [50] Referring to triangle MCD in fig.2, let the following definitions apply: a = length of side MC = r b = length of side MD = c = length of side CD = r C = angle at vertex M = π ⁄ 2 -Δ Δ = heading error relative to e -dge of protection field, Δ ‹ π ⁄ 2 for an approaching missile [51] Applying the cosine law and simplifying the term COS(A+B) as COS(A)COS(B)- SIN(A)SIN(B) yields the following expression: where: r = minimum missile turn radius r′ = radial range to target (m) d = minimum acceptable missile miss distance, measured from ship centre (m) [52] After isolating ∆ in eqn (12a), the heading error φ, relative to the sensor in the centre of the protection field, required for the missile to pass through point D in fig.2 (edge of the protection field) is given by the following expression: eqn (12b) [53] Substituting eqn (12b) into the expression for bearing rate for a given heading error relative to the sensor location (missile’s target) (eqn (7)) yields: eqn (13) [54] Using the identity of eqn (9), eqn (13) can be rewritten as: eqn (14) [55] The arctangent terms in eqn (14) can be rewritten as arcsine and arccosine form as shown below: eqn (15) eqn (16) [56] Substituting eqns (15) and (16) into eqn (14) yields the following expression for the rate of change of missile bearing for a marginal miss of the protection field: eqn (17) [57] The arcsine term in eqn (17) can be rewritten as: eqn (18) [58] Substituting eqn (18) into eqn (17) eliminates the arcsine and cosine terms to yield: eqn (19) [59] Eqn (19) can be simplified further by moving in the denominator of the second term in the brackets into its numerator, to create a common denominator of 2 ^^: eqn (20) [60] Eqn (20) can be written as the sum of two terms; substituting the expression for the missile’s minimum turn radius r (eqn (4)) yields the final form of the expression for the bearing rate required for a marginal miss of the protection field: eqn (21) [61] The first term of the right hand side of eqn (21) is independent of range between the missile and the target, and is identically the expression for required bearing rate for zero protection field width (eqn (11)). The second term of eqn (21) is dependent on range between the missile and the target, and the protection field width parameter d. Considering the positive root, the second term of eqn (21) is a correction term which increases the required bearing rate according to the protection field width variable d. [62] The range dependence of the second term of eqn (21) means that the exact bearing rate required for soft kill, for a given protection field width, can be determined if the missile’s range is tracked. Unfortunately, in practice it may be difficult or impossible to accurately track a missile during an engagement, depending on available engagement time lines, tracking radar cueing and tasking, tracking radar characteristics, and missile altitude and radar cross section, for example. However, if the correction term is small, then the required bearing rate may be dominated by the first term of eqn (21), and it may be possible to assess whether the missile has been defeated by the countermeasures without range tracking the missile, i.e. it may be possible to confidently confirm soft kill based solely on observed bearing rate. [63] A strategy for using eqn (21) to assess soft kill without range tracking the missile might take the following form: if a missile range is inferred from knowledge of the missile deployment strategy, known or inferred characteristics, and the elapsed time of the terminal phase of the engagement, then the instantaneous bearing rate observed during the engagement can be translated via eqn (21) into an inferred kill range based on protection field width. For example, if the inferred kill range is less than or equal to the inferred range of the missile, then it may be possible to calculate the probability of successful soft kill based on the probability that the true missile range is greater than the kill range for the observed bearing rate. That probability may change during the engagement as the observed bearing rate changes, until soft kill is a certainty. [64] The size of the correction term of eqn (21) can be quantified for illustration purposes, using a missile velocity of 300 m/sec and an agility of 2.5 gravities. The required bearing rate for soft kill as a function of range is presented in fig. 4, for several protection field widths. At a range of 3000 m, the bearing rate which must be observed for soft kill is between 2.7 and 3.2 degrees/sec, considering a protection field width of +/-200 to +/-500 m. This range is narrowed to between 2.5 and 2.7 degrees/sec if the protection field width is limited to +/-100 to +/-200 m. [65] The size of the normalized bearing rate correction term of eqn (21) relative to the fixed term is presented as a function of range in fig. 5, for several protection field widths. For a protection field width of +/-200 m, the correction term is less than 20% at a range of 3000 m, and approximately 4.2% at 5000 m. This suggests that in cases where a missile is outside the range of close-in weapons systems (e.g.3000 m), soft kill can be confidently confirmed without range tracking the missile. [66] Concepts described here could possibly be applied to the problem of third party protection against missile attack. It may be possible to derive mathematical relationships defining conditions under which intercept of other nearby ships is impossible, based on (a) engagement and scenario parameters observable from the jammer ship and (b) knowledge of the location of a third party ship. [67] Under the approach described here, it is theoretically possible to relate observed bearing rate to closest possible approach of the missile to the target. The relationship is expressed in eqn (21), where bearing rate is known and eqn (21) is solved for the protection field width d. The closest approach d for a given bearing rate thus has a dependency on range and bearing rate which is weak at long range, and becomes stronger at close range. This relationship could theoretically be used to estimate worst-case missile miss distance continuously during an engagement, allowing a dynamic definition of soft kill dependent on the engagement scenario. For example, if there were a number of inbound missiles and time management of ECM resources between all missiles was critical, it might be best for ship survival if soft kill were defined using a modest protection field width. This would result in an earlier assessment of soft kill than if soft kill defined using a larger protection field width, and ECM resources could be more expeditiously directed to the next priority threat. Definition of protection field width could be managed in real time by an expert system, in order to maximize ship survivability of the primary targeted ship and any nearby ships as well. Definition of missile intercept corridor [68] The concepts described above can be used to define a subset of points in three-dimensional space defined by an ordered triple (x,y,z), from which is it is physically possible for the missile to intercept the target, given the current heading error. The subset of points from which it is physically possible for the missile to intercept the target is here referred to as the missile intercept corridor. The missile intercept corridor is calculated by applying the current missile velocity vector to all points in space, calculating the unique missile heading error for each point, and using eqn (5) to test for the ability of the missile to execute a course correction to intercept the protection field. If the required course correction does not exceed the limitations of missile agility, then that point in space is inside the missile intercept corridor. If the required course correction exceeds the limitations of missile agility, then the point is outside the missile intercept corridor. [69] Considering a missile which is guided in both azimuth and elevation, limitations of missile agility result in constraints on dive and climb angle as well as azimuth heading in order to ensure that an intercept is possible. In the case of a dive, the missile must be able to recover before impacting the sea or land surface. In the case of a climb, the missile must be able to recover before overflying the target. The set of points is found to represent a funnel shape, loosely speaking. Considering a missile which is guided in the azimuth plane only, as are some sea-skimming missiles, the missile intercept corridor collapses to a funnel shape impressed on the sea surface, as shown qualitatively in figs. 6A and 6B for the case of zero heading error. The arrows in fig. 6A represent missile heading. Figs. 6C- 6E show alternative views of a funnel-shaped missile intercept corridor for a missile guided in both azimuth and elevation, including an isometric view (Fig. 6C), a ship view (Fig. 6D) and a top down view (Fig.6E). [70] Because eqn (5) is a function of heading error φ, and because heading error at any point in space depends on the current missile velocity vector, the shape of the missile intercept corridor is a function of the missile velocity vector. This means that the shape of the missile intercept corridor dynamically changes as the missile velocity vector changes during missile flight. Intuitively, if the missile yaws in a given direction, the missile intercept corridor yaws in the opposite direction as seen by the missile. If the missile yaws left, the points above the sea surface from which the missile can intercept the target are displaced to the right as seen by the missile. A qualitative visualization of the shape of the missile intercept corridor is shown in fig. 7, for the case of a sea-skimming missile and nonzero heading error. The arrows in this figure represent missile heading. [71] Eqn (12b) expresses the relationship between missile radial range and the minimum heading error required to cause the missile miss the target by the specified protection field width. [72] Eqn (5) can be generalized to account for the fact that the missile must fail to intercept the ground or sea surface, i.e. it must be able to recover from an ECM-induced dive. The missile intercept corridor forms a surface which spreads out over the sea or ground, defined by points in space from which ground intercept is unavoidable given the current missile heading, velocity and agility. The mathematical definition of this surface is similar to that given by eqn (5), except the missile must remain inside the missile intercept corridor in order to effect soft kill. [73] The most immediate value of the concept of a missile intercept corridor is that it gives quantitative meaning to what is sometimes referred to as soft kill. Soft kill is defined as an instance of missile engagement in which the missile fails to pass within the protection field. If the missile crosses the missile intercept corridor boundary, even momentarily, it cannot by definition of the missile intercept corridor, re-enter the corridor and consequently, by definition of the missile intercept corridor, cannot pass through the protection field. Logically, every missile which fails to pass through the protection field (i.e. misses the target) must cross the missile intercept corridor boundary at some point in its flight, either early or late in the engagement. If the miss distance is small, then the missile crosses the missile intercept corridor at near range, late in the engagement. If the miss distance is large, it crosses the boundary at long range. Visualizations of these two cases are presented in figs. 8A and 8B, considering an offboard decoy as the countermeasure. [74] The missile intercept corridor can also theoretically be used as a dynamic measure of countermeasure effectiveness; if the bearing rate is increasing, the countermeasure is causing the missile to approach the missile intercept corridor boundary. It is also possible that adaptive, closed loop control of ECM could be based on the missile intercept corridor concepts and the use of bearing rate as a measure of effectiveness (MOE) for the countermeasure. Under such a scheme, observed changes in bearing rate could be correlated with changes in countermeasures deployment to adaptively tune countermeasures deployment to cause confirmed soft kill. [75] Perhaps most importantly, the concepts and mathematical formulations presented here could be used to determine the moment at which an onboard jammer should be turned off. The flow of reliable guidance information to the missile becomes extremely critical as the missile approaches the edge of the corridor; if this information flow is interrupted, such as by shutting off an active onboard jammer or somehow forcing the missile into reacquisition, then it might cross the missile intercept corridor boundary. [76] It may be possible to use the concept of a missile intercept corridor to define different mechanisms of soft kill as caused by active onboard angle deception countermeasures. Some possible examples are: (i) Information Starvation: Soft kill results from shutting the jammer off or otherwise forcing reacquisition when the missile is near the missile intercept corridor boundary; (ii) Catastrophic Maneuver: The countermeasure causes a radical trajectory maneuver when the missile is near the missile intercept corridor boundary; (iii) Cumulative Deviation: The countermeasure causes a series of gradually accumulating small trajectory deviations until the missile impinges on the missile intercept corridor. [77] The design, ranking, and even real-time selection of onboard and offboard countermeasures could be based on different mechanisms of soft kill, derived from intercept corridor related concepts, in order to maximize the probability of missile soft kill. Anticipatory Jammer Turn-Off [78] The missile intercept corridor concepts can be used to cause missile soft kill by information starvation by turning off the jammer when crossing of the missile intercept corridor is immanent. Jammer turn-off is controlled by bearing rate and acceleration conditions, measured at the jammer, based on missile intercept corridor formulations. This feature of jammer operation is here referred to as anticipatory jammer turn-off (AJTO). [79] The geometry for deriving the bearing rate and acceleration conditions for AJTO is presented in fig. 9. Fig. 9 (not to scale) illustrates the missile flight trajectory for sufficiently short interval that the flight trajectory is well approximated by a straight line. The missile crosses the missile intercept corridor boundary at point M, at time t = 0.0 sec. The bearing rate of the missile at the missile intercept corridor boundary, as seen by the jammer, is given by expressions derived previously (eqn (11) for a point target, eqn (21) for a distributed target of projected width 2d). The objective of the AJTO analysis is to define bearing rate and acceleration conditions at point M', where crossing of the missile intercept corridor is immanent, in the absence of correcting guidance information. If the jammer is turned off when the missile is at point M' and the missile is denied correcting guidance information, it will cross the missile intercept corridor boundary and subsequent recovery will be impossible. Assuming that the missile velocity is constant and the flight trajectory is well approximated by a straight line, the point M' corresponds to a time-to-go t for the missile to reach the missile intercept corridor boundary. The requirement that the flight trajectory be well approximated by a straight line means that placement of point M' (choice of t) can be related to the expected time constant of the overall missile guidance loop. [80] The missile flight trajectory in the vicinity of point M is approximated by a straight line: eqn (22) where the slope m is given by the following expression: eqn (23) where φ is the missile heading error relative to the jammer at the missile intercept corridor boundary. The y and z components of missile position as shown in fig.9 are functions of time, and can be written as shown below: eqn (24) [81] The look angle θ from the jammer to the missile position at AJTO (point M) is given by: eqn (25) [82] The bearing rate of the missile at any point on the line given by eqn (22) can be expressed by differentiating eqn (25) with respect to time: eqn (26) [83] Eqn (26) can be rewritten with a common denominator as shown below: eqn (27a) [84] Note that at t = 0 eqn (27a) reduces to eqn (7), as expected. Eqn (27a) can be rewritten by expanding the numerator and denominator products: eqn (27b) [85] The heading error φ of the missile at the missile intercept corridor boundary is given by eqn (6) for the point target case, and by eqn (12b) for the distributed target case. For simplicity, eqn (6) is now substituted into eqn (27b) in order to derive the missile bearing rate at point M' in terms of parameters which are knowable and/or measurable at/by the jammer. The AJTO equations in this derivation now commit to zero protection field width, though a separate derivation using eqn (12b) for the distributed target case is easily done. [86] Referring again to eqn (6), let the following assignments apply: eqn (6) eqn (28) [87] Eqn (27b) can thus be rewritten as shown below, using substitutions comprising eqn (28): eqn (29) [88] The following trigonometric identities can be used to simplify eqn (29): eqn (30) [89] Using eqn (30), the following expressions are derived: eqn (31) eqn (32) [90] The minus signs in eqn (32) have been cancelled because sine is an odd function (i.e. sin(-x) = -sin(x)). [91] Arccosine can be converted to arcsine as follows: eqn (33) [92] Substituting eqn (33) into eqn (32) yields: eqn (34) [93] Substituting eqns (31) and (34) into eqn (29) yields:

eqn (35) [94] Eqn (35) reduces to the following expression, for t = 0: eqn (36) [95] Eqn (36) is consistent with eqn (11), i.e. bearing rate for zero protection field width. Note that the solution of eqn (35) can be either complex or real, depending on the square root term in the denominator. The solution is completely real providing the following inequality is satisfied: eqn (37) [96] The bearing rate for jammer RF turn-off (eqn (35)) is completely real only if the radial range between the jammer sensor and the missile is less than twice the missile minimum turn radius. [97] Interestingly, the solution to eqn (35) can in principle be converted from complex to real at a given range by increasing the closing velocity v _m between the missile and the jammer sensor. Although missile velocity cannot be affected by the jammer, it is possible to increase v _m by heading toward the missile; if the targeted platform heads directly toward the missile, the value of v _m becomes the sum of the missile velocity and the platform velocity. [98] Eqn (35) may have a tactical application in airborne applications in which a maneuvering or maneuverable aircraft is under attack by an anti-air missile. For example, if the protection field is sufficiently small, as might be the case for a physically small target such as an aircraft, then depending on the user’s requirements the simplest version of the soft kill equation may provide a good enough assessment of the soft kill state of the missile. i.e. there might be negligible benefit in including the correction term for the physical width of the protection field. The soft kill equation could also or instead be used to plan defensive maneuvers by an aircraft under attack by an anti-air missile. For example, this could involve either the point target version of the equation (eqn 36) or distributed target versions of the equation (eqn 35). A defensive plan is possible because the shape and position of the intercept corridor can be determined by missile’s closing speed, its maximum lateral acceleration, and the velocity vector of the missile relative to the aircraft velocity vector. Since the closing speed and relative velocity vector can be changed by intentional aircraft maneuvers, it may be possible in theory to create an aircraft flight plan to cause the missile to leave the intercept corridor for example as soon as possible, or at a predetermined time and location in 3D-space. Which equation is used depends on the trade-off between processing speed and the user’s requirements for confidence and accuracy in determining whether or when missile soft kill has occurred. [99] Eqn (35) gives the bearing rate of the missile as seen by the jammer t seconds before the missile crosses the missile intercept corridor boundary for zero protection field width, assuming that the missile flies a straight line trajectory. Note that at time t = 0 the missile crosses the missile intercept corridor boundary; t must be negative in eqn (35) to give the bearing rate before the missile crosses the missile intercept corridor boundary. [100] Eqn (35) assumes that the missile flies in a straight line tangential to a circle which just barely touches an edge of the protection field. In this condition, the missile has by definition reached the intercept corridor boundary. Looking backward in time from this point, if the time step is sufficiently small then the missile flight trajectory can be approximated by a straight line. Although the missile flight path is not generally known, the straight-line approximation is best if the missile really is flying in a straight line, and worst of the missile is in a max g turn. If the straight line flight hypothesis is valid or sufficiently valid, then eqn (35) can be used to calculate the observed missile bearing rate when the missile is t sec from crossing the intercept corridor boundary. If t is smaller than the latency of the missile’s guidance system to correct a heading error, then it may be possible to cause a missile miss by for example turning off the jammer at this moment, t sec before the missile reaches the boundary. If the jammer is unintentionally providing guidance information to the missile, then turning off the jammer will deny the missile guidance information when it is close to the Intercept Corridor boundary. [101] Eqn (35) therefore determines the bearing rate at which AJTO should be applied, i.e. when the jammer should be turned off. How successful this is in a given engagement may depend on the choice of t, and the real-world validity of the assumption that the missile flight path is well-approximated by a straight line over t seconds. [102] A typical calculation can be used to illustrate the application of eqn (35), and give a quantitative comparison of bearing rate at soft kill with bearing rate at AJTO. Consider the following parameters: v _m = 300 m/s a _max = 2.0 gravities t = -2.0 sec r′ = 4000 m [103] According to eqn (35), the missile bearing rate as seen by the jammer 2 sec. before it crosses the missile intercept corridor boundary, assuming no correction of flight trajectory, is 1.449 degrees/sec. If the missile were to cross the missile intercept corridor boundary, the observed bearing rate would be 1.874 degrees/sec (eqn (11)). Note that because eqn (11) is formulated for zero protection field width, there is no range dependence on bearing rate at the missile intercept corridor, though there is range dependence in eqn (35). According to the inequality of eqn (37), the maximum range at which there is a real solution to eqn (35) is 9.174 km. If the ship velocity is 20 kt (37 kph, 10.3 m/sec) and the ship heading is aligned with the inbound missile, this range could be increased by 622 m to 9796 m. [104] In some implementations, the choice of time delay t may be determined taking into account various engagement parameters/guidance properties, such as overall missile guidance loop time constant, likely reacquisition time, scenario specifics, maximum lateral acceleration, desired soft kill range, desired miss distance statistics, etc. Application to Multiple Targets Case [105] Missile intercept corridor concepts and mathematical formulations can be applied to engagements where there is more than one potential target or emitter present. In this case, the probability of missile soft kill (P _sk ) is inherently associated with the identity of an emitter, because a missile flight trajectory which comprises a hit for one emitter (missile successfully intercepts emitter), comprises a miss for all other emitters in the scenario. If there are multiple emitters in an engagement scenario then the P _sk for a single missile must be expressed for each emitter. This means that an emitter identity is implicit in the missile intercept corridor equation. [106] The application of the concept of a missile intercept corridor to multiple ship platforms is shown qualitatively in fig. 10. During a missile attack in which two or more ships are present, it is possible to classify whether soft kill has occurred for each ship, when soft kill occurred, and possibly to identify critical points in the engagement where soft kill transitions happen - or can happen - for each ship. Non-limiting examples of uses for this information could include: (a) selecting and/or controlling the deployment of countermeasures to maximize the probability of survival for any or all ships, (b) preventing misallocation of electronic warfare (EW) resources to a missile which is no longer a threat, (c) in the event of a stream or barrage attack by multiple missiles, prioritizing the missiles in terms of the threat they pose to individual ships, or (d) developing a rule-based system for self- or group-deployment of countermeasures for self or group defense, including for example the following elements: (i) the time-to-impact for each ship for which soft kill has not yet occurred, (ii) the quantity and location of available EW resources, (iii) an automatically generated, automatically coordinated and automatically executed plan for self- or group-defense, including 1) a schedule of soft kill and/or hard kill for each missile, 2) the selection and scheduling of soft kill and/or hard kill weapon deployment, 3) the real-time sharing of information about actions taken by each ship for coordination confirmation purposes, and 4) the real-time sharing of real-time observations of the missile attack viewed from multiple observation points to create a confirmed tactical picture against which the defense plan acts. (iv) real-time effectiveness assessment so the self- or group-defense plan can be adaptive in real-time according to measured success, or lack of success, at each step in the defense plan. [107] Consider a missile attack against a fleet of ships. It is reasonable to assume that the effectiveness of a countermeasure deployed by one ship in the fleet is dependent on (a) the actions of other member ships in the fleet, and (b) the ship which is initially targeted by the missile. This means that optimization of the effectiveness of ship defense against missile attack when there is more than one ship present thus involves selecting countermeasures and tactics to optimize a survival matrix, where dimensions of the matrix are determined by the separate or combination conditions under which the P _sk of each ship is to be calculated. The elements of the matrix are conditional probabilities. Although the matrix may be multi-dimensional, it can be illustrated in two dimensions by considering P _sk for each of three ships given that a particular one of the three is initially targeted by the missile. The form of the matrix is shown below: Table 1: Example Conditional Probability of Survival Matrix Other Soft Kill Definitions [108] Although the present disclosure has focused primarily on soft kill defined by a heading error which is physically unrecoverable, based on the missile’s aerodynamic design (e.g., missile agility), the embodiments described herein could also or instead relate to definitions of soft kill based on heading errors which are functionally unrecoverable, rather than physically unrecoverable, from the perspective of an aerial or nautical weapon’s other design features, e.g., the specific design of the missile seeker and/or autopilot. For example, this elaboration may be relevant in instances in which a missile develops a large heading error at a long range, and while the missile may be physically agile enough to intercept the target (e.g., the range may be greater than the diameter of a circle inscribed by a maximum-g turn for the missile), the engagement may be outside the missile’s design envelope (e.g., in principle a missile could fly in a tightest-possible circle and depart). [109] Several non-limiting examples of possible criteria for defining a functionally unrecoverable heading error are presented below, for illustration. Untrackable Target [110] If the target is outside the field of view of the seeker such that the target cannot be acquired by the seeker. In this case soft kill has been successful because suitable guidance cannot be restored, even though the missile may be physically agile enough to correct its heading error. Ill-Conditioned Guidance Loop [111] Consider the case in which the seeker angle tracks the primary target using a sidelobe track point. Although such tracking may provide target angle information, the associated sight line rate estimate may be uncalibrated for a Proportional Navigation autopilot designed to work with the boresight copolar descriminator slope. In this case, soft kill may be defined as successful, depending on the interplay of several other variables. Target Rejection [112] Consider a missile designed to reject decoys. If the ship can produce a response in the seeker which causes the seeker to classify the ship as an invalid target, soft kill has been successful. Alternate Target Selection [113] Consider a missile designed to monitor two or more potential targets, and to home on the highest priority one. If an induced heading error can cause the missile to alter its target priority and home on a different target, soft kill has been successful with respect to the initial primary target. Zero/Low Probability of Intercept [114] It may be possible to define a heading error for one or more specific missile designs, such that the probability of target reacquisition and/or target intercept is effectively zero. It is reasonable to assume that some or all missiles might incorporate an own-platform safety feature by which the missile self-destructs if it develops an absolute heading greater than a threshold value. [115] This option addresses cases in which the missile develops a maximum-g turn at long range, i.e. at a range greater than the diameter of a maximum-g turn. Excessive Flight Time [116] If the missile flight time required to intercept the target exceeds the design limitations, then soft kill has occurred. This possibility logically covers the case of a maximum-g turn at long range, i.e. outside the diameter of a maximum-g turn; if the missile is required to fly a complete maximum-g circuit, or execute a maximum-g zig-zag before re-engaging the target, the flight distance may exceed the missile design specifications. This criterion is illustrated in figure 11. Example Apparatus [117] Figure 12 illustrates an example of an apparatus for assessing ability of a maneuverable first object to intercept an area or volume containing a second object, according to an embodiment of the present disclosure. The apparatus includes an engagement monitoring and/or management system 100 operably connected to an ECM system 110. The ECM system 110 is operably connected to a bearing measurement device 112 and a range measurement device 114. [118] In some embodiments, the engagement monitoring and/or management system 100 may be configured to determine, based on guidance properties of a first object relative to an area or volume containing a second object, whether the first object is capable of intercepting the area or volume containing the second object. In such embodiments, the engagement monitoring and/or management system 100 may be further configured to produce an indication of a result of the determination. In other embodiments, the engagement monitoring and/or management system 100 may also or instead be configured to use the guidance properties of the first object to determine an intercept corridor defining a subset of points in three-dimensional space from which is it is physically possible for the first object to intercept the area or volume containing the second object. In such embodiments, the engagement monitoring and/or management system 100 may be further configured to produce an indication of the intercept corridor. For example, as shown in figure 12, the engagement monitoring and/or management system 100 may include a soft kill calculator 102 and a soft kill result display 104, wherein the soft kill calculator 102 is configured to determine whether the first object is capable of intercepting the area or volume containing the second object and/or to determine an intercept corridor and produce an output that enables the soft kill result display 104 to display an indication of the result. [119] In some cases, the guidance properties include at least: an observed rate of bearing change of the first object relative to the area or volume containing the second object; speed and/or velocity of the first object; and a maximum lateral acceleration limit for the first object. In some cases, the guidance properties may also or instead include at least: an observed heading error of the first object relative to the area or volume containing the second object; speed and/or velocity of the first object; and the maximum lateral acceleration limit for the first object. [120] The ECM system 110 is configured to radiate signals with the intention to render a hostile weapon system ineffective. For example, the ECM system 110 may include an active jamming system onboard the second object. For example, the ECM system 110 may be an ECM system configured to counter missile systems. In some embodiments, the ECM system 110 may further include at least one active or passive offboard decoy located away from the second object. In some embodiments, the ECM system 110 is configured to track the bearing and/or range of an object using the bearing measurement device 112 and/or the range measurement device 114. [121] In some embodiments, the engagement monitoring and/or management system 100 is configured to obtain or estimate the guidance properties of the first object based on information received from the ECM system 110. For example, in some embodiments the soft kill calculator 102 may be configured to obtain or estimate the guidance properties based on bearing information and/or range information gathered by the ECM system 110 using the bearing measurement device 112 and/or the range measurement device 114. [122] In some embodiments, the engagement monitoring and/or management system 100 is configured to obtain or estimate guidance properties for the first object over time and update the intercept corridor over time using the updated guidance properties. In some cases, the engagement monitoring and/or management system 100 is configured to update the intercept corridor in real time or near real time as updated guidance properties, such as updated velocity data, are obtained during flight of the first object. [123] In some embodiments, the engagement monitoring and/or management system 100 is configured to adapt the ECM system 110 based on proximity of the first object to a boundary of the intercept corridor. For example, the engagement monitoring and/or management system 100 may be configured to turn off RF transmission from at least one jamming system of the ECM system 110 t seconds in advance of a time at which the first object is anticipated to cross an intercept corridor boundary, as described earlier. [124] In some embodiments, the engagement monitoring and/or management system 100 may be partially or wholly integrated with the ECM system 110. For example, in some embodiments the soft kill calculator 102 may be integrated with the ECM system 110 in the form of software stored in memory forming part of the ECM system, and run by a processor forming part of the ECM system, possibly with some hardware additional hardware and/or firmware. [125] In general, hardware, firmware, components which execute software, or some combination thereof, might be used in implementing the functionality of the engagement monitoring and/or management system 100 and/or the ECM system 110. Electronic devices that might be suitable for implementing such functionality include, among others, microprocessors, microcontrollers, Programmable Logic Devices (PLDs), Field Programmable Gate Arrays (FPGAs), Application Specific Integrated Circuits (ASICs), and other types of “intelligent” integrated circuits. Software could be stored in memory for execution. The memory could include one or more physical memory devices, including any of various types of solid-state memory devices and/or memory devices with movable or even removable storage media. Conclusion [126] The present disclosure has defined a quantitative meaning for the term soft kill by introducing the concept of a guaranteed missile miss. This concept is based on limitations of missile agility and heading errors induced by angle deception ECM. The angle deception may be caused by either onboard or offboard countermeasures. The concept of a protection field width has been introduced as part of the definition of soft kill, i.e. soft kill is defined as an instance of missile engagement in which the missile fails to pass within the protection field, notionally centered on the target. [127] Mathematical relationships have been described which define the range at which a given heading error is unrecoverable, for the missile to pass through the protection field. Additionally, a mathematical relationship has been defined from which the observed rate of change of missile bearing is related to the range at which the missile is unable to intercept the protection field. If the protection field collapses to a point, the bearing rate required for a missile miss is independent of missile range. The bearing rates for soft kill are expected to be of the order of 2-3 degrees/sec, and are assessed to be measurable using existing technology. The concept of anticipatory jammer turn-off (AJTO) has been introduced, and expressions have been derived to control jammer turn-off based on observed missile bearing rate and acceleration in order to cause the missile to cross the missile intercept corridor boundary. [128] The concept of soft kill has been shown to imply an identity in the sense that soft kill means that a particular target or platform survived a missile attack. In cases where there is more than one potential target for the missile, a different probability of soft kill may be calculated for each target using the missile intercept corridor equations. Optimization of the survival of a set of targets requires optimization of conditional probabilities in a survivability matrix. [129] It may be possible to apply the concepts and mathematical formulations in the following ways: (i) to dynamically assess missile soft kill in real time during a missile engagement, (ii) to form the basis of adaptive closed-loop control of an active onboard jammer, (iii) to determine the best moment to turn off an active onboard jammer, (iv) to confirm soft kill for a protection of a third party ship, (v) to calculate the closest possible approach of the missile based on real-time observable engagement parameters, e.g., determine the value of d (fig.2) which satisfies eqn. (21) for the observed bearing rate and missile range, (vi) to apply mathematical relationships related to confirmation of soft kill through an expert system to enhance ship survivability by real-time engagement management and management and conservation of ship borne softkill and hardkill resources, (vii) to define different mechanisms of soft kill, (viii) select and/or control the deployment of countermeasures to maximize the probability of survival for any or all ships, (ix) prevent misallocation of EW resources to a missile which is no longer a threat, (x) in the event of a stream or barrage attack by multiple missiles, prioritize the missiles in terms of the threat they pose to individual ships, or (xi) develop a rule-based system for self- or group-deployment of countermeasures for self or group defense, including for example the following elements: (a) the time-to-impact for each ship for which soft kill has not yet occurred, (b) the quantity and location of available EW resources, (c) an automatically generated, automatically coordinated and automatically executed plan for self- or group-defense, including 1) a schedule of soft kill and/or hard kill for each missile, 2) the selection and scheduling of soft kill and/or hard kill weapon deployment, 3) the real-time sharing of information about actions taken by each ship for coordination confirmation purposes, and 4) the real-time sharing of real-time observations of the missile attack viewed from multiple observation points to create a confirmed tactical picture against which the defense plan acts. (d) real-time effectiveness assessment so the self- or group-defense plan can be adaptive in real-time according to measured success, or lack of success, at each step in the defense plan. [130] Although the present disclosure has focused primarily on soft kill defined by a heading error which is physically unrecoverable, based on the missile’s aerodynamic design, the embodiments described herein could also or instead relate to definitions of soft kill based on heading errors which are functionally unrecoverable, from the perspective of an aerial or nautical weapon’s other design features. This elaboration may be relevant in instances in which a missile develops a large heading error at long range, and while the missile may be physically agile enough to intercept the target, the engagement may be outside its design envelope, for example. [131] Furthermore, although many of the examples described above relate primarily to protection of ships from aerial attack by anti-ship missiles (ASMs), the concepts underlying the embodiments described herein could also or instead be applied to other types of protection scenarios, such as the protection of ships from nautical weapons (e.g., torpedoes), or protection of aircraft from aerial weapons (e.g., air-to-air missiles or surface-to-air missiles) in air-to-air and/or surface-to-air attack scenarios. [132] The inventive concepts disclosed herein are not limited in their application to the details of construction and the arrangement of the components set forth in the description or illustrated in the drawings. The inventive concepts disclosed herein are capable of other embodiments or of being practiced or carried out in various ways. Also, it is to be understood that the phraseology and terminology employed herein is for the purpose of description and should not be regarded as limiting the inventive concepts disclosed and claimed herein in any way. [133] Numerous specific details are set forth in order to provide a more thorough understanding of the inventive concepts. However, it will be apparent to one of ordinary skill in the art that the inventive concepts within the instant disclosure may be practiced without these specific details. In other instances, well-known features have not been described in detail to avoid unnecessarily complicating the instant disclosure. [134] As used herein, the terms "comprises," "comprising," "includes," "including," "has," "having" or any other variation thereof, are intended to cover a nonexclusive inclusion. For example, a composition, a process, method, article, or apparatus that comprises a list of elements is not necessarily limited to only those elements but may include other elements not expressly listed or inherently present therein. [135] Unless expressly stated to the contrary, "or" refers to an inclusive or and not to an exclusive or. For example, a condition A or B is satisfied by anyone of the following: A is true (or present) and B is false (or not present), A is false (or not present) and B is true (or present), and both A and B are true (or present). An inclusive or may be understood as being the equivalent to: at least one of condition A or B. [136] In addition, use of the "a" or "an" are employed to describe elements and components of the embodiments herein. This is done merely for convenience and to give a general sense of the inventive concepts. This description should be read to include one or at least one and the singular also includes the plural unless it is obvious that it is meant otherwise. [137] Any reference to "one embodiment" or "an embodiment" means that a particular element, feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment. The appearances of the phrase "in one embodiment" in various places in the specification are not necessarily all referring to the same embodiment. [138] Further, use of the term “plurality” is meant to convey “more than one” unless expressly stated to the contrary. [139] Circuitry, as used herein, may be analog and/or digital, components, or one or more suitably programmed microprocessors and associated hardware and software, or hardwired logic. Also, "components" may perform one or more functions. The term "component," may include hardware, such as a processor, an application specific integrated circuit (ASIC), or a field programmable gate array (FPGA), or a combination of hardware and software. Software includes one or more computer executable instructions that when executed by one or more component cause the component to perform a specified function. It should be understood that the algorithms described herein are stored on one or more non-transitory memory. Exemplary non-transitory memory includes random access memory, read only memory, flash memory or the like. Such non-transitory memory may be electrically based or optically based. [140] As used herein the terms "approximately," "about," "substantially" and variations thereof are intended to include not only the exact value qualified by the term, but to also include some slight deviations therefrom, such as deviations caused by measuring error, manufacturing tolerances, wear and tear on components or structures, stress exerted on structures, and combinations thereof, for example. For example, as used herein, the term “substantially” means that the subsequently described parameter, event, or circumstance completely occurs or that the subsequently described parameter, event, or circumstance occurs to a great extent or degree. For example, the term “substantially” means that the subsequently described parameter, event, or circumstance occurs at least 90% of the time, or at least 91%, or at least 92%, or at least 93%, or at least 94%, or at least 95%, or at least 96%, or at least 97%, or at least 98%, or at least 99%, of the time, or means that the dimension or measurement is within at least 90%, or at least 91%, or at least 92%, or at least 93%, or at least 94%, or at least 95%, or at least 96%, or at least 97%, or at least 98%, or at least 99%, of the referenced dimension or measurement.