Filtered Modified MITRI (FMM) Risk Index in LEO

• The paper refines MITRI by concurrently updating dynamic components (local crowding, debris generation, and collision probability) during simulation, thereby capturing latent debris risks.
• FMM is defined as a debris-risk index that integrates a mass scaling of M^(1.75) with dynamically updated parameters, enhancing the identification of high-risk objects for active debris removal.
• Implemented within the MOCAT-MC framework, FMM demonstrates superior performance over static indices by accurately ranking high-criticality debris, even under varied update cadences.

Searching arXiv for the specified FMM papers and closely related uses of the acronym. Filtered Modified MITRI (FMM) is a debris-risk index for Low Earth Orbit (LEO) designed to prioritize targets for active debris removal (ADR) by extending MIT’s MITRI index with modified dynamic risk terms and filtering rules for practically relevant debris. In the formulation reported by the authors, FMM preserves the product structure of MITRI but changes how the dynamic components are computed: local crowding $R$, yearly generated debris $D$, and collision probability $P$ are updated concurrently during Monte Carlo simulation, and the debris-generation term is based on fictitious collisions rather than only realized collisions. The index was developed and validated within the MOCAT-MC simulation framework, with the stated goal of improving identification of high-criticality debris while retaining a physically grounded dependence on mass, orbital lifetime, and environmental density (Medhin et al., 21 Jul 2025).

1. Origin and problem setting

FMM was introduced in the context of orbital sustainability in an increasingly congested LEO environment, where collisions between large derelicts can generate thousands of fragments and contribute to a self-sustaining collision cascade. Within this setting, ADR is treated as a resource-constrained intervention: each mission can deorbit only a small number of objects per year, so target ranking becomes a central technical problem. The paper positions FMM as a response to that ranking problem by refining a pre-existing dynamic index, MITRI, rather than replacing the broader risk-index paradigm altogether (Medhin et al., 21 Jul 2025).

The starting point is the observation that established indices such as CSI, $R_N$, and ECOB are largely static or “offline,” in the sense that they are computed from a single environmental snapshot and do not fully account for the evolving, stochastic character of the debris population. MITRI had already moved beyond that by combining static terms with dynamic “online” terms derived from MOCAT-MC. FMM is presented as a further refinement motivated by two stated limitations of original MITRI: dynamic terms were computed post-simulation via distribution fitting rather than updated concurrently, and the yearly debris term $D$ counted only realized collisions, thereby missing latent risk from high-probability conjunctions that did not happen to materialize in a given Monte Carlo realization (Medhin et al., 21 Jul 2025).

2. Mathematical structure and defining modifications

The paper retains the outer multiplicative structure of MITRI. In that structure, the principal factors are the mass term $\left(\frac{M}{M_0}\right)^{1.75}$, the background-density and inclination factor $\rho_B(h) g(i)$, the normalized residual lifetime $\frac{L}{L_0}$, and expected normalized dynamic terms for local density $R$, debris generation $D$, and collision probability $D$0 (Medhin et al., 21 Jul 2025).

What makes the index “Modified” and “Filtered” is not a new outer formula but a change in the internal definitions of the dynamic terms. The authors describe two major modifications. First, $D$1, $D$2, and $D$3 are computed concurrently during simulation using a smoothed average rather than ex-post statistical fitting. Second, $D$4 is redefined using fictitious collisions at every time step, so that potential debris generation is accumulated even when a conjunction does not become a realized collision in a particular Monte Carlo run (Medhin et al., 21 Jul 2025).

The “Filtered” aspect concerns which fragments are counted in the debris-generation term. Only fragments above practical detection or relevance thresholds are included: mass > 10 kg or characteristic length > 10 cm. The paper states that this aligns the index with standard “trackable debris” definitions and suppresses noise from very small fragments with less operational relevance. In parallel, background density is periodically recomputed so that the environmental term remains synchronized with the evolving population (Medhin et al., 21 Jul 2025).

The concurrent expectation update is given as

$D$5

for a dynamic quantity $D$6 such as $D$7, $D$8, or $D$9. This makes FMM online in the authors’ sense: at any time step, the score reflects time-averaged behavior up to that point rather than a statistic derived only after a full simulation horizon has been completed (Medhin et al., 21 Jul 2025).

Dynamic background density is likewise updated over time through

$P$0

where $P$1 is the number of objects in altitude shell $P$2 at time $P$3, $P$4 is the shell volume, the shell width is 50 km, and recomputation is performed every 1, 3, or 6 months over a 200-year horizon (Medhin et al., 21 Jul 2025).

3. Physical basis of the mass term

A central feature of FMM is the mass exponent

$P$5

The paper explicitly grounds this exponent in two physical contributions: a probability component of approximately $P$6, associated with the tendency of more massive objects to have larger cross-sectional area and hence higher collision probability, and a consequence component of approximately $P$7, derived from the NASA Standard Breakup Model (SBM), where catastrophic fragment counts scale approximately as $P$8 (Medhin et al., 21 Jul 2025).

The cited catastrophic and non-catastrophic fragment-count relations are

$P$9

and

$R_N$0

respectively. In the paper’s interpretation, mass therefore enters risk assessment twice: through collision probability and through fragment-generation potential (Medhin et al., 21 Jul 2025).

This physical grounding is not merely formal. The reported sensitivity analysis shows that removing the mass term produces what the authors call catastrophic performance degradation, with even 5 removals per year yielding worse final debris populations than baseline. A linear mass term, $R_N$1, also underperforms the $R_N$2 formulation for both MITRI and FMM. The stated conclusion is that mass is indispensable and that the $R_N$3 exponent is not arbitrary but anchored in fragmentation physics (Medhin et al., 21 Jul 2025).

4. Computation within MOCAT-MC

FMM is implemented inside MOCAT-MC, an open-source Monte Carlo LEO debris-evolution model developed by MIT’s Aerospace Robotics and Control Lab. The framework propagates objects individually, includes drag and $R_N$4, simulates station-keeping and evasive maneuvers for active satellites, models collisions via NASA EVOLVE 4 and SBM, includes explosions with empirical failure rates, samples projected launch activity, and represents post-mission disposal successes and failures (Medhin et al., 21 Jul 2025).

For each Resident Space Object, the required inputs include mass, cross-sectional area, and orbital elements such as semi-major axis or altitude and inclination. These feed the derivation of residual lifetime $R_N$5 through drag decay, while $R_N$6, $R_N$7, and $R_N$8 are computed through the CUBE method and breakup physics. Space is discretized into 3D CUBEs and altitude shells; local crowding is estimated cube-wise; collision probabilities are computed among objects in the same cube; and fictitious debris generation is accumulated for each potential conjunction pair, subject to the 10 kg / 10 cm filter (Medhin et al., 21 Jul 2025).

The pairwise collision probability used in the framework is reported as

$R_N$9

with total collision probability in a cube expressed as

$D$0

under an independence assumption across pairwise events. Within FMM, the time-averaged $D$1 is updated concurrently by the smoothed expectation formula rather than reconstructed after the simulation (Medhin et al., 21 Jul 2025).

The paper emphasizes that FMM is more computationally expensive than MITRI because it processes all potential collision pairs at each time step rather than only rare realized events. The reported runtime increase is approximately 1.54× for a static FMM run relative to MITRI and up to approximately 1.7× when background updates are frequent. That overhead is partly offset by algorithmic choices such as the CUBE method, an analytical propagator, and concurrent running averages rather than post-hoc statistical fitting (Medhin et al., 21 Jul 2025).

5. Empirical performance and sensitivity

The principal benchmark reported for FMM is the High-Risk Identification Rate, defined as the percentage of “ground truth” high-collision objects—objects experiencing $D$2 collision events in any of 1000 Monte Carlo seeds—that are ranked within the top 0.5% by the index (Medhin et al., 21 Jul 2025).

Background density update	MITRI	FMM
Static (1 time)	89.90%	98.15%
Every 6 months	84.04%	98.13%
Every 3 months	83.33%	99.99%
Every 1 month	87.37%	98.08%

These results show FMM at approximately 98–100% identification across update cadences, whereas MITRI ranges from approximately 83–90%. The paper further reports that FMM ranks true high-risk objects more critically than CSI and clusters them within the top 0.5% of rankings, whereas CSI yields a broader distribution (Medhin et al., 21 Jul 2025).

Several sensitivity studies are reported. Introducing probability-weighted fictitious-debris scaling through two $D$3 models worsened performance for all removal rates $D$4, increasing final object counts by approximately 0.4–7%. Raising the debris filter threshold from >10 kg to >50 kg produced almost identical performance, though slightly worse with the higher threshold. More frequent background-density updates improved or preserved FMM performance, whereas MITRI performance degraded under the same change. The authors interpret these results as evidence that the core FMM architecture—fictitious collisions, dynamic background density, and $D$5—is already close to optimal within the simulation context (Medhin et al., 21 Jul 2025).

6. Operational interpretation, limits, and terminological disambiguation

Operationally, FMM prioritizes objects that are massive, long-lived, situated in dense orbital regions, and implicated in many high-risk conjunctions even if no realized collision occurred in a specific run. The paper reports that risk-driven removal by MITRI or FMM substantially outperforms random removal, and specifically that removing 1 high-MITRI object per year outperforms randomly removing 5 objects per year in stabilizing the LEO population. A more nuanced result concerns cadence: for annual removals, MITRI yields lower final debris populations than FMM, whereas for 5-year and 10-year cadences FMM performs marginally better, suggesting that FMM’s accumulated potential-risk perspective is more advantageous when interventions are infrequent (Medhin et al., 21 Jul 2025).

The stated limitations are correspondingly specific. FMM is model-dependent, relying on SBM, the CUBE methodology, and MOCAT-MC assumptions about launches and disposal. It is tailored to LEO rather than MEO or GEO. It requires reasonably accurate mass and cross-sectional estimates, which are not always available. It is also computationally heavier than MITRI because fictitious collision processing expands the dynamic workload (Medhin et al., 21 Jul 2025).

The acronym itself requires disambiguation. In two unrelated arXiv papers on ECG anomaly detection and neuronal signal clustering, FMM does not mean Filtered Modified MITRI; it denotes Frequency Modulated Möbius, a family of nonlinear parametric oscillatory models used for ECG morphology and spike waveform analysis (Verardo et al., 2023, Rodríguez-Collado et al., 2022). Those works explicitly state that the phrase “Filtered Modified MITRI” does not appear there. A plausible implication is that acronym overlap can create bibliographic ambiguity, but the underlying models, application domains, and mathematical objects are entirely different: Filtered Modified MITRI is a risk index for ADR in LEO, whereas Frequency Modulated Möbius is a parametric waveform model for oscillatory biological signals (Medhin et al., 21 Jul 2025).