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Minimum orbit intersection distance
Minimum orbit intersection distance
Minimum orbit intersection distance
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The orbit of (4953) 1990 MU, which, with a MOID of 0.0263 AU, is classified as a potentially hazardous object

Minimum orbit intersection distance (MOID) is a measure used in astronomy to assess potential close approaches and collision risks between astronomical objects.[1][2] It is defined as the distance between the closest points of the osculating orbits of two bodies. Of greatest interest is the risk of a collision with Earth. Earth MOID is often listed on comet and asteroid databases such as the JPL Small-Body Database. MOID values are also defined with respect to other bodies as well: Jupiter MOID, Venus MOID and so on.

An object is classified as a potentially hazardous object (PHO) – that is, posing a possible risk to Earth – if, among other conditions, its Earth MOID is less than 0.05 AU. For more massive bodies than Earth, there is a potentially notable close approach with a larger MOID; for instance, Jupiter MOIDs less than 1 AU are considered noteworthy since Jupiter is the most massive planet.[1]

A low MOID does not mean that a collision is inevitable as the planets frequently perturb the orbit of small bodies. It is also necessary that the two bodies reach that point in their orbits at the same time before the smaller body is perturbed into a different orbit with a different MOID value. Two objects gravitationally locked in orbital resonance may never approach one another. Numerical integrations become increasingly divergent as trajectories are projected further forward in time, especially beyond times where the smaller body is repeatedly perturbed by other planets. MOID has the convenience that it is obtained directly from the orbital elements of the body and no numerical integration into the future is used.[3]

The only object that has ever been rated at 4 on the Torino Scale (since downgraded), the Aten asteroid (99942) Apophis, has an Earth MOID of 0.00026 AU (39,000 km; 24,000 mi). This is not the smallest Earth MOID in the catalogues; many bodies with a small Earth MOID are not classed as PHO's because the objects are less than roughly 140 meters in diameter (or absolute magnitude, H > 22). Earth MOID values are generally more practical for asteroids less than 140 meters in diameter as those asteroids are very dim and often have a short observation arc with a poorly determined orbit. As of September 2023, there have been seven objects detected and their Earth-MOID calculated before the Earth impact.[4] The first two objects that were detected and had their Earth-MOID calculated before Earth impact were the small asteroids 2008 TC3 and 2014 AA. 2014 AA is listed with a MOID of 0.00000045 AU (67 km; 42 mi),[5] and is the second smallest MOID calculated for an Apollo asteroid after 2020 QY2 with an Earth-MOID of 0.00000039 AU (58 km; 36 mi).[6]

Potentially hazardous asteroids with Earth MOID < 0.0004 AU (~60,000 km or ~5 Earth diameters) include:[7]
Object Earth MOID
(AU)		 Size (m)
(approximate)		 (H)
2016 FG60 0.000076 AU (11,400 km; 7,100 mi)[8] 300 21.1
(177049) 2003 EE16 0.000107 AU (16,000 km; 9,900 mi) 320 19.8
2012 HZ33 0.000131 AU (19,600 km; 12,200 mi) 260 20.4
2010 JE88 0.000148 AU (22,100 km; 13,800 mi) 180 21.5
(137108) 1999 AN10 0.000153 AU (22,900 km; 14,200 mi) 1300 17.9
2022 BX1 0.000177 AU (26,500 km; 16,500 mi) 170 21.7
2003 EG16 0.000179 AU (26,800 km; 16,600 mi) 490 19.4
2021 NQ5 0.000187 AU (28,000 km; 17,400 mi) 210 21.2
(442037) 2010 PR66 0.000238 AU (35,600 km; 22,100 mi) 695 19.3
(216985) 2000 QK130 0.000252 AU (37,700 km; 23,400 mi) 200 21.3
99942 Apophis 0.000257 AU (38,400 km; 23,900 mi) 370 19.7
(89958) 2002 LY45 0.000261 AU (39,000 km; 24,300 mi) 1300 17.2
(35396) 1997 XF11 0.000305 AU (45,600 km; 28,400 mi) 704 17.0
162173 Ryugu 0.000315 AU (47,100 km; 29,300 mi) 896 19.6
(143651) 2003 QO104 0.000321 AU (48,000 km; 29,800 mi) 2300 16.1
(85236) 1993 KH 0.000335 AU (50,100 km; 31,100 mi) 500 18.8
(471240) 2011 BT15 0.000368 AU (55,100 km; 34,200 mi) 150 21.4
Numbered periodic comets with Earth MOID < 0.02 AU (~3 million km) include:
Object Epoch Earth MOID
(AU)
3D/Biela 1832 0.0005 AU (75,000 km; 46,000 mi; 0.19 LD)
109P/Swift-Tuttle 1995 0.0009 AU (130,000 km; 84,000 mi; 0.35 LD)
55P/Tempel–Tuttle 1998 0.0085 AU (1,270,000 km; 790,000 mi; 3.3 LD)
255P/Levy 2007 0.0088 AU (1,320,000 km; 820,000 mi; 3.4 LD)
15P/Finlay 2015 0.0092 AU (1,380,000 km; 860,000 mi; 3.6 LD)
73P–BW 2022 0.0093 AU (1,390,000 km; 860,000 mi; 3.6 LD)[9]
252P/LINEAR 2016 0.0122 AU (1,830,000 km; 1,130,000 mi; 4.7 LD)
460P/PanSTARRS 2016 0.0163 AU (2,440,000 km; 1,520,000 mi; 6.3 LD)
289P/Blanpain 2019 0.0165 AU (2,470,000 km; 1,530,000 mi; 6.4 LD)
21P/Giacobini–Zinner 2017 0.0179 AU (2,680,000 km; 1,660,000 mi; 7.0 LD)
Some well known Main-belt asteroids
with Earth MOID < 1 AU
Object Earth MOID
(Asteróide[10])
6 Hebe 0.975 AU (145.9 million km; 90.6 million mi; 379 LD)
7 Iris 0.850 AU (127.2 million km; 79.0 million mi; 331 LD)
8 Flora 0.873 AU (130.6 million km; 81.2 million mi; 340 LD)
12 Victoria 0.824 AU (123.3 million km; 76.6 million mi; 321 LD)
18 Melpomene 0.811 AU (121.3 million km; 75.4 million mi; 316 LD)
84 Klio 0.798 AU (119.4 million km; 74.2 million mi; 311 LD)
228 Agathe 0.657 AU (98.3 million km; 61.1 million mi; 256 LD)

See also

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References

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Minimum orbit intersection distance

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The Minimum Orbit Intersection Distance (MOID) is the shortest distance between the closest points of the osculating orbits of two celestial bodies, such as asteroids and planets or satellites and , calculated in three-dimensional at a given using Keplerian . This metric quantifies the geometric proximity of orbital paths without accounting for relative velocities or perturbations, making it a fundamental tool for assessing collision hazards in near-Earth space. MOID plays a central role in the classification and monitoring of near-Earth objects (NEOs), particularly in identifying potentially hazardous asteroids (PHAs), which are defined as NEOs with an MOID to Earth of 0.05 astronomical units (au) or less—equivalent to approximately 7.5 million kilometers—and an absolute magnitude (H) of 22.0 or brighter, indicating objects roughly 140 meters or larger in diameter assuming typical albedos. For instance, this threshold helps prioritize surveillance efforts by NASA's Center for Near-Earth Object Studies (CNEOS), filtering thousands of tracked objects to focus on those posing realistic impact threats over extended periods. Beyond asteroids, MOID is applied to space debris analysis, where it aids in evaluating risks to operational satellites by prefiltering large catalogs for potential conjunctions. Computing MOID accurately is computationally intensive due to the nonlinear of elliptical orbits, often requiring solutions to systems of equations derived from like semi-major axis, eccentricity, and inclination. Traditional algebraic methods solve for critical points where the line connecting the bodies is perpendicular to both velocity vectors, while numerical iterative approaches, such as those using or , enhance efficiency for real-time applications. More recent asymptotic techniques expand solutions in terms of small eccentricities or critical anomalies, achieving sub-meter precision with up to 40% faster computation times compared to exact methods, as demonstrated in analyses of NEO catalogs like NEODyS. Ongoing advancements as of 2025 include non-Keplerian methods for perturbed orbits and improved conjunction filters. These developments are crucial for processing vast datasets from observatories, enabling proactive mitigation strategies like deflection missions for high-risk objects.

Definition and Background

Definition

The minimum orbit intersection distance (MOID) is defined as the minimum between any two points on the osculating Keplerian orbits of two celestial bodies, assuming unperturbed motion around a common . This geometric measure represents the closest possible approach between the orbital paths, computed using the instantaneous at a specific without considering gravitational perturbations. Osculating orbits approximate the true trajectories as ellipses (or other conics) at a given time, providing a snapshot of the bodies' positions under idealized two-body dynamics. Unlike the actual close approach distance, which accounts for the relative timing, positions, and of the bodies at a specific moment, MOID focuses solely on the spatial of the orbits and ignores whether the bodies occupy those closest points simultaneously. This distinction makes MOID a conservative indicator of potential rather than a predictor of real-time encounters, serving as an initial filter in . For instance, a low MOID suggests orbital paths that cross or nearly cross, but a collision requires additional alignment in time and . MOID is typically expressed in astronomical units (AU) or kilometers, with values below certain thresholds indicating heightened concern; for example, near-Earth asteroids with an Earth MOID of less than 0.05 AU are classified as potentially hazardous asteroids (PHAs) if their absolute magnitude is 22.0 or brighter. Conceptually, the MOID occurs at points where the line connecting the two bodies is perpendicular to both tangent vectors (i.e., velocity directions) of the orbits, ensuring the distance is locally minimized along the common normal. This perpendicularity condition arises from the geometry of curve-to-curve distances in three-dimensional space.

Historical Context

The concept of the minimum orbit intersection distance (MOID) emerged in the mid-1990s amid growing concerns over impact hazards, as astronomers sought efficient metrics to prioritize near-Earth objects (NEOs) for follow-up observations. It was formally introduced by Edward Bowell and Karri Muinonen in 1994, who defined MOID as the smallest separation between two Keplerian orbits and proposed its use to classify potentially hazardous s (PHAs) with an MOID below 0.05 AU combined with an brighter than H=22. This threshold reflected the scale at which planetary perturbations could significantly alter an 's trajectory toward , marking a shift from qualitative risk assessments to quantitative orbital geometry analysis. By the late 1990s, MOID had been adopted into operational databases for NEO monitoring, including NASA's (JPL) Small-Body Database and the International Astronomical Union's (MPC), enabling systematic risk screening as discovery rates surged. This integration coincided with the formalization of PHA designations by the MPC around 1999, where MOID served as a primary filter for objects warranting detailed impact probability studies. The 2000s brought milestones in MOID application through expanded NEO surveys, such as the (LINEAR) program, operational since 1998, and NASA's Near-Earth Asteroid Tracking (NEAT) survey, which collectively discovered thousands of objects and necessitated routine MOID computations to manage data overload. These efforts transformed MOID from a theoretical tool into a standard metric for cataloging orbital risks. In the , refinements accelerated with data from the European Space Agency's mission, launched in 2013, whose precise reduced uncertainties in . Key publications laid the groundwork for these developments, including Bowell and Muinonen's seminal 1994 chapter and A. Carusi and E. Dotto's 1996 analysis in Icarus on close encounters of minor bodies with . These works, published in high-impact venues like the Press proceedings and Icarus, emphasized geometric intersections over full dynamical simulations, influencing subsequent astrodynamics research.

Mathematical Formulation

Orbital Parameters Involved

The computation of the minimum orbit intersection distance (MOID) relies on the classical Keplerian to define the shape, size, orientation, and position of each orbit involved. These elements include the semi-major axis (a), which determines the overall scale of the orbit; the eccentricity (e), which describes its shape from circular to highly elliptical; the inclination (i), which specifies the tilt of the relative to the reference plane; the (Ω), which locates the point where the orbit crosses the reference plane ascending; and the argument of periapsis (ω), which measures the angle from the ascending node to the periapsis. Each of these five elements must be specified for both orbits to enable the of their intersection. The standard reference frame for MOID calculations is the heliocentric , centered on the Sun with the plane (Earth's orbital plane) as the fundamental plane and the vernal equinox defining the zero direction. This frame ensures consistency for solar system bodies, such as asteroids and planets. When orbital data is provided in geocentric coordinates—for instance, for Earth-orbiting objects or initial tracking—transformations to the heliocentric frame are required, typically involving vector additions of Earth's position and velocity relative to the Sun to derive the equivalent heliocentric elements. For precise MOID determination, instantaneous osculating elements are employed, representing the Keplerian ellipse that best fits the orbit at a specific moment by matching position and instantaneously. These elements are computed at a common for both bodies to reflect their contemporaneous configurations, avoiding discrepancies from temporal mismatches. In contrast, elements, which average out short-period perturbations like those from Earth's oblateness, are generally unsuitable for MOID as they do not capture the exact instantaneous needed for minimum distance assessment. A special case arises when the orbits are coplanar, with zero relative inclination after alignment to the reference frame, simplifying the to the minimum distance between points on the two conics within that plane. This configuration eliminates out-of-plane components, allowing the distance to be evaluated solely through in-plane separations, often using parametric representations of the ellipses.

Computation Algorithms

The computation of the minimum orbit intersection distance (MOID) relies on a geometric approach that minimizes the between position vectors on two confocal Keplerian orbits. Specifically, the MOID is found by solving for the true anomalies f1f_1 and f2f_2 that minimize d=∥r1(f1)−r2(f2)∥d = \|\mathbf{r}_1(f_1) - \mathbf{r}_2(f_2)\|, where r1\mathbf{r}_1 and r2\mathbf{r}_2 are the position vectors of the primary and secondary orbits, respectively. This minimum occurs at points where the relative position vector r1−r2\mathbf{r}_1 - \mathbf{r}_2 is to both orbital velocity vectors v1\mathbf{v}_1 and v2\mathbf{v}_2, leading to the constraints (r1−r2)⋅v1=0(\mathbf{r}_1 - \mathbf{r}_2) \cdot \mathbf{v}_1 = 0 and (r1−r2)⋅v2=0(\mathbf{r}_1 - \mathbf{r}_2) \cdot \mathbf{v}_2 = 0. A key analytical formulation involves reducing the problem to a quartic equation in the eccentric anomalies, derived from the relative motion equations analogous to Lagrange's planetary equations. This quartic arises when expressing the distance minimization in the orbital plane, often after separating in-plane and out-of-plane components, and solving the stationary points of the squared distance function. For non-coplanar orbits, the full system may require higher-degree polynomials, such as a 16th-degree algebraic equation obtained by trigonometric substitutions in the eccentric anomaly domain. Numerical methods are essential for solving these non-linear equations, particularly when analytical solutions are infeasible. Iterative solvers, such as the Newton-Raphson method or (which offers cubic convergence), are commonly employed to find roots of the distance function or equations, typically requiring fewer than 2 iterations per root for convergence. For highly eccentric orbits, asymptotic approximations provide efficient alternatives; for instance, series expansions in the critical eccentric anomaly up to order N=2N=2 yield MOID values with errors below 6.4×10−116.4 \times 10^{-11} AU (approximately 9.6 meters) while reducing computation time by 40% compared to exact methods. These expansions, integrated into the Space Dynamics Group (SDG) MOID algorithm, use coefficients like u∗(N)(a,e,α,β)=∑i=0Nci(a,α,β)eiu_*^{(N)}(a,e,\alpha,\beta) = \sum_{i=0}^{N} c_i(a,\alpha,\beta) e^i, where aa is the semi-major axis ratio, ee the eccentricity, and α,β\alpha, \beta angular parameters defining relative orientation. Software implementations of these algorithms enable efficient MOID calculations for large catalogs, achieving O(1) per orbital pair through direct optimization rather than full propagation. Open-source libraries like distlink, a C++ tool based on algebraic polynomial root-finding with error controls via Newton iterations and discrete Fourier transforms, support elliptic orbits and handle degenerate cases like near-circular or coplanar configurations with high reliability (computation times of 24–44 µs per pair on standard hardware). NASA's Center for Studies (CNEOS) employs similar geometric and iterative methods for MOID assessments in planetary defense tools, though data from the HORIZONS serves as input for such computations. Precision is limited by , but robust implementations achieve accuracies better than 10−1010^{-10} AU, with error estimation via root uncertainty propagation ensuring safe handling of near-collision scenarios.

Applications

Planetary Defense

In planetary defense, the minimum orbit intersection distance (MOID) serves as a critical metric for identifying near-Earth objects (NEOs) that pose potential collision threats to , enabling early detection and prioritization of hazardous asteroids. By calculating the closest possible approach between an asteroid's orbit and , MOID helps filter objects from NEO surveys for further risk assessment, focusing resources on those with trajectories that could intersect Earth's path. This application is essential for mitigating impacts from objects larger than approximately 140 meters, which could cause regional devastation. Potentially hazardous asteroids (PHAs) are defined as NEOs with an Earth MOID of 0.05 AU or less and an H of 22.0 or brighter, corresponding to diameters roughly 140 or larger. This threshold identifies objects capable of close approaches that, under orbital perturbations, might lead to impacts. For instance, 99942 Apophis, classified as a PHA with an MOID of approximately 0.0003 AU, exemplifies high-risk cases; its 2029 flyby at about 31,000 km from prompted extensive monitoring to refine impact probabilities over the next century. MOID integrates with impact probability assessments in systems like NASA's Sentry, which scans catalogs to detect potential impacts up to 100 years ahead, using low-MOID objects as initial candidates for detailed propagation and probability calculations via the and Torino scales. Objects with MOID values below 0.05 AU are prioritized on the Sentry Risk Table if their orbits indicate non-zero impact chances, allowing rapid filtering from surveys like those from the Catalina Sky Survey. Case studies highlight MOID's role in threat monitoring and mitigation planning. Asteroid (612901) 2004 XP14, a PHA with an MOID of about 0.0013 AU, underwent intensified radar observations during its 2006 close approach at 0.00289 AU, informing models for future passes, including monitoring for potential risks around 2027. Similarly, in the Double Asteroid Redirection Test (DART) mission, the binary system 65803 Didymos (MOID ≈ 0.04 AU) was selected as a deflection target due to its accessible near-Earth orbit, allowing kinetic impact testing without actual hazard, to validate technologies for altering PHA trajectories. The follow-up Hera mission, launched in 2024, will study the Didymos system post-DART to assess deflection effectiveness, further leveraging MOID for planetary defense validation as of 2025. Monitoring programs leverage MOID for global coordination, with NASA's Center for Near-Earth Object Studies (CNEOS) and the European Space Agency's Near-Earth Object Coordination Centre (NEOCC) collaborating to rank risks. Both centers use MOID thresholds to populate shared risk lists, such as the Sentry Risk Table and ESA's Risk List, ensuring synchronized prioritization of PHAs for follow-up observations and deflection studies through the International Asteroid Warning Network.

Space Mission Planning

In space mission planning, the minimum orbit intersection distance (MOID) plays a pivotal role in designing and analyzing orbits for artificial satellites and interplanetary probes, enabling the assessment of potential close approaches and the implementation of safety protocols. By computing the MOID between a spacecraft's trajectory and those of other objects, mission planners can identify risks early in the design phase and adjust parameters to maintain operational safety. This metric is particularly valuable in low Earth orbit (LEO) environments, where dense populations of satellites and debris necessitate precise conjunction screening. Avoidance maneuvers rely heavily on MOID calculations to ensure safe passage between and or planetary bodies. For instance, in LEO operations, conjunction assessments target collision probabilities below acceptable thresholds, such as 10^{-4} per event, using tools that propagate to predict minimum distances over short propagation windows, typically 5 days. In one validated analysis of over 272,000 resident space object pairs, non-Keplerian MOID tools achieved 99% accuracy within 1 km, directly supporting maneuver decisions by distinguishing true risks from conservative estimates. MOID also contributes to in multi-body environments, particularly through extensions of that incorporate . In such designs, the spacecraft's is evaluated for its closest approach to assisting bodies to determine the altitude of periapsis, ensuring efficient energy gain while avoiding unintended intersections. For example, during the Voyager missions, which employed multiple and Saturn , trajectory planning refined flyby geometries using calculations to balance scientific objectives with safe periapsis distances, typically on the order of planetary radii plus atmospheric margins. This approach allows planners to iterate on patched conic approximations, minimizing fuel while verifying non-intersection with other orbital paths. For debris field analysis, MOID is applied to cataloged space junk using environmental models to generate collision avoidance alerts. The European Space Agency's (ESA) MASTER model simulates debris populations in LEO and beyond, providing input for MOID computations across thousands of objects to prioritize high-risk conjunctions. In operational settings, this supports systems like ESA's Space Debris Office services, which issue alerts based on MOID thresholds (e.g., below 30 km for initial screening) and propagate uncertainties to forecast maneuver needs, reducing false positives in busy orbital regimes. Such analyses have been to missions like Sentinel satellites, where debris-induced MOID evaluations inform routine avoidance campaigns. In future missions, MOID ensures safe lunar orbit insertions by verifying distances to Earth-orbiting assets exceed established thresholds. For the , trajectory planning incorporates MOID evaluations during and (NRHO) establishment to avoid conjunctions with LEO constellations and debris, drawing from precedents like the ARTEMIS mission's collision avoidance operations with Earth-orbiting probes. These assessments, often using covariance propagation, target miss distances aligned with guidelines (e.g., probability-based thresholds of 10^{-4}), safeguarding multi-element architectures involving the .

Limitations and Advanced Considerations

Non-Keplerian Influences

Real-world orbital dynamics deviate from the idealized Keplerian assumptions underlying standard MOID calculations due to various non-Keplerian influences, including gravitational perturbations from other bodies, non-gravitational forces, relativistic effects, and full multi-body interactions. These factors cause the actual minimum distance between orbits to vary over time, potentially leading to close approaches that differ significantly from two-body predictions. While basic Keplerian MOID provides a static snapshot, incorporating these influences is essential for accurate long-term in planetary defense. Gravitational perturbations from major planets, particularly , induce secular variations in orbits that directly affect MOID values. Jupiter's influence drives quasi-linear trends in MOID evolution accompanied by periodic oscillations, with flybys causing abrupt changes in such as eccentricity and inclination. For near-Earth asteroids (NEAs), approximately 13.27% exhibit MOID variations exceeding 0.001 AU over 200-year timescales due to these perturbations, while 5.88% show changes greater than 0.01 AU, especially for objects with aphelia beyond 4 AU. For instance, the NEA 2012 TJ146 experiences notable MOID shifts from Jupiter encounters, highlighting how such perturbations can alter collision probabilities over decades. Non-gravitational forces, such as the Yarkovsky effect, further modify small orbits by inducing gradual semimajor axis drift through asymmetric . This effect is most pronounced for asteroids smaller than 20 km in diameter, causing drift rates of up to 2 × 10^{-5} AU per million years for 5-km objects and higher for meter-sized bodies, on the order of AU per million years. Over collisional , these drifts can shift orbits by up to 0.1 AU for small asteroids, facilitating transitions into resonant configurations that alter MOID with or other planets dynamically. Observations of NEAs like 6489 Golevka demonstrate detectable semimajor axis changes accumulating over multiple apparitions, impacting MOID assessments for potential close approaches. Relativistic corrections from introduce subtle perturbations to orbital motion, primarily through perihelion precession, which are negligible for most MOID calculations but become relevant for precise modeling in the inner solar system. At 1 AU, these effects amount to about 10^{-8} in normalized units, with perihelion advances of 0.0384 arcseconds per revolution, scaled by (1 - e^2) for eccentric orbits. For high-eccentricity inner-system asteroids like 1566 (e ≈ 0.83), the annual precession reaches 0.101 arcseconds, accumulating to several arcseconds over decades and reducing positional residuals by up to 30% when included in models. Such corrections ensure accuracy for Mercury-crossing objects where two-body approximations fail, though they rarely alter overall MOID by more than a few kilometers for typical NEAs. In multi-body environments, transitioning from two-body MOID to n-body minimum distances requires to account for cumulative perturbations from all solar system bodies. Tools like N-body code enable this by simulating collisional dynamics with symplectic integrators and algorithms, such as octree-based searches that identify close approaches scaling efficiently with particle number. These simulations compute time-evolving minimum distances by tracking full trajectories, revealing intersections or near-misses that static Keplerian methods overlook, particularly in dense regions like the main .

Uncertainty and Propagation

Observational uncertainties in , arising from measurement errors in and ranging, introduce significant ambiguity in the computation of the minimum orbit intersection distance (MOID). These errors are typically represented by matrices that quantify the statistical spread in the estimated orbital parameters, such as semi-major axis, eccentricity, and inclination. Propagating these through the MOID calculation allows for the estimation of regions around the nominal MOID value, essential for assessing collision risks in planetary defense scenarios. Covariance propagation for MOID involves linearizing the distance function between two confocal Keplerian orbits around the nominal orbital elements and applying the covariance matrix to derive the variance of the MOID. For two orbits with nominal elements Eˉ1\bar{\mathbf{E}}_1 and Eˉ2\bar{\mathbf{E}}_2, the covariance matrix ΓEˉ\Gamma_{\bar{\mathbf{E}}} is block-diagonal, and the propagated uncertainty in a regularized MOID map d~h\tilde{d}_h is given by Γd~h(Eˉ)=(∂d~h∂E(Eˉ))ΓEˉ(∂d~h∂E(Eˉ))T\Gamma_{\tilde{d}_h}(\bar{\mathbf{E}}) = \left( \frac{\partial \tilde{d}_h}{\partial \mathbf{E}} (\bar{\mathbf{E}}) \right) \Gamma_{\bar{\mathbf{E}}} \left( \frac{\partial \tilde{d}_h}{\partial \mathbf{E}} (\bar{\mathbf{E}}) \right)^T, yielding a standard deviation σh=Γd~h(Eˉ)\sigma_h = \sqrt{\Gamma_{\tilde{d}_h}(\bar{\mathbf{E}})}
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