Physics-Informed Buffers & MTEBB Envelope

Reading the scene

The terrain block is a synthetic 72 km square with the facility at its centre. The solid white ring is the statutory perimeter and the hatched orange band is the shadow zone, shaded by its equation (4) weight. Each modality drapes its footprint, the ground from which it reaches the facility: gold where a sight line from the facility observer lands, blue with outward ripples where an RF link closes, green where the road service area reaches, and violet where a UAS launch is feasible. The roads shade from bright to dark green as the drive from the facility lengthens and turn grey beyond T_net. Two dashed rings are limits, not tests: the gold viewshed analysis radius and the violet UAS transit range. The thin red line outlines the envelope, the union of the enabled footprints, so it follows terrain and roads rather than tracing a circle. Rings beyond the terrain continue on the dark base plate, which is how the 100-mile military perimeter remains visible.

Towns, streets, lots, and buildings come into focus as the camera closes in. Each lot is tinted by its zone and its buildings carry a lighter tint of the same colour: white when core, orange in the shadow zone, red when several modalities reach it, a modality colour when exactly one does, magenta when only the minimum rule excludes it, and grey outside the universe. From a distance a town reads as one patch of colour; at street level each lot carries its own. A beacon on a leader line marks each modality's farthest reach, the point that sets its R. The lettered pins mark specimen locations, placed once per scene where a single rule decides: just past the statutory line, at the shadow edge, behind a ridge, off the road, in a radio shadow, under a blocked drone command link. Their chips are coloured by current zone, so a control change shows at once which specimens it moves. Clicking a specimen, or anywhere else, places a test parcel that reports every pathway test with the sight line, RF loss terms, the drive from the facility, and the command-link result behind it.

What the model calculates

The scene

From a seed: a 125 m heightfield, towns on low and gentle ground, a road network routed around steep ground and water with a junction wherever two roads meet, and lots along every road. Each lot is a rectangle whose front line is the edge of the road's right-of-way, holding a main building and, where they fit, accessory buildings. A parcel is tested at its main building's centre, at ground level.

Four tests per parcel

VIEW: does a straight line from the facility observer reach the parcel's ground over the terrain, inside the analysis radius? EM: does the path loss from a 3 m antenna on the parcel to a 10 m antenna at the facility fit the EM rating's budget? NET: over every road point within the access distance, what is the earliest arrival from the facility, and is it inside T_net? UAS: is the parcel on land, within the platform group's transit range, outside prohibited airspace, and, for a piloted mission, in sight of the aircraft over the facility? Each answer is yes or no, and the evidence behind it is kept for the test parcel's card.

Three pathways and the zone

Pathway 1 admits a parcel within the statutory distance; Pathway 3 within α_shadow times it, recording the equation (4) weight; Pathway 2 where any enabled modality says yes under the maximum rule, or every one does under the minimum. A parcel admitted by any pathway enters 𝒰 in the first zone that applies: core, then shadow, then physics-extended.

The ranges and equation (1)

Each modality's R is the farthest point it reaches: the farthest cell of its footprint or the farthest parcel it admits, whichever is farther, and for NET the farthest road point reached within T_net plus the access distance. Equation (1), MTEBB_f = max(R_view, R_em, R_net, R_uas), is therefore a true bound: no lot that Pathway 2 admits lies beyond it. It is reported as the extent of the envelope and never used as a circle that admits parcels. A test parcel clicked between the drapes' samples can pass a test a little beyond them; lots in the buffers are admitted however far they lie. Where a road reached within T_net leaves the scene, the service area continues past the scene's edge, so R_net there is a lower bound on the true reach, and the panel says so.

Drawn and tested

The drapes evaluate the same physics on the terrain grid, for context: a 2,048-ray viewshed sweep, a 720-ray RF field, the service area on 125 m cells, and the command-link viewshed from the aircraft. Admission, counts, and lot colours come only from the per-parcel tests, so a lot at the edge of a drape can differ from the colour beneath it.

What updates

A control change recomputes the affected footprints, every parcel's tests, the ranges, equation (1), the zone counts, and the specimens' zones. A new terrain preset or seed regenerates the scene: terrain, roads, lots, and buildings.

Three pathways into 𝒰

Pathway 1 · Euclidean buffer

The 31 CFR Part 802 distances: one mile around standard critical infrastructure and one hundred miles around designated military installations. The perimeter is a legally operative bright line, and because it is published, an acquirer can sit at 1.02 miles and retain full physical capability. The sandbox keeps the statutory circle and places that evasion parcel on request.

Pathway 2 · Physics-informed MTEBB

Each modality admits a parcel only where its physics reaches the facility from that parcel. VIEW requires a clear sight line from a 10 m facility observer to the parcel's ground inside a 10 km analysis radius, with Earth curvature and refraction. EM requires the RF link to close: free-space loss, knife-edge diffraction over the dominant terrain obstruction, and clutter must fit the budget of the facility's EM rating, so ridges cast radio shadows and higher bands reach less far. NET requires road access: a road point within the access distance must be reached from the facility within 30 minutes over the drawn road network. UAS requires a feasible launch: on land within the transit range of the platform group matched to the facility's susceptibility, outside prohibited airspace, and, for a piloted mission, with command-and-control line of sight from the operator on the parcel to the aircraft over the facility. Each modality's R is the distance to the farthest point of its footprint or of the lots it admits, and equation (1), max(R_view, R_em, R_net, R_uas), is the bounding extent of their union: no lot that Pathway 2 admits lies beyond it, and inside it only the tests admit. The maximum rule is the union, so one modality suffices. The minimum, the intersection, excludes every parcel that some modality reaches and another does not, and the sandbox counts and marks exactly those when it is selected. The methodology's 0, 15, and 50 km R_em placeholders are kept for comparison: the panel counts the parcels inside a placeholder whose link fails and those beyond it whose link closes, the re-evaluations EM_PLACEHOLDER_QUALIFIED anticipates.

Pathway 3 · Shadow buffer

The statutory distance extended by α_shadow, provisionally 1.5, with influence falling linearly from 1 at the statutory boundary to 0 at the outer edge. Admission stays binary at the outer edge, so the cliff an acquirer would exploit moves from one mile to one and a half rather than disappearing; the weight grades each parcel's influence for the stages that follow. For the 100-mile perimeter the 150-mile edge outruns any optical-surveillance rationale, which is why facility-type multipliers are the recommended resolution beyond the pilot.

Specification

Question

Which parcels could physically enable an effect against a protected facility, beyond the reach of a fixed statutory radius?

Method

Three qualification pathways: the 31 CFR Part 802 statutory buffer; the physics-informed MTEBB, in which each modality admits a parcel only where it reaches the facility (a clear sight line from a 10 m observer inside a 10 km radius, an RF link whose free-space, knife-edge diffraction, and clutter loss fits the EM-rating budget, road access inside a 30-minute service area on the drawn road network, and a UAS launch within the platform-group range with command-link line of sight), with equation (1) the bounding extent of what they reach; and an α = 1.5 shadow buffer with linear distance decay. Any single pathway admits a parcel.

Data

Procedural 72 × 72 km terrain at 125 m in three presets; a planar road network of highways, arterials, rural lanes, and town street grids, routed within class grade limits wherever the 250 m lattice allows (ridge crossings exceed them, about 6 to 14% of highway length on the ridge preset); and about 16,000 to 20,000 street-fronting lots with 28,000 to 35,000 buildings (houses, mixed-use, office buildings, warehouses, and farmsteads) at typical small-town dimensions and setbacks; declared illustrative, never a pilot result.

Implementation

React and Three.js; modality footprints, rings, and the envelope outline drawn in the terrain shader so they drape at any zoom; lots drawn with their boundaries in a fragment shader and buildings as instanced meshes at true size; scene generation in about 1.5 to 2.5 s, viewshed in about 20 ms, RF field in about 0.1 s, and every parcel re-tested and re-admitted on each control change; up to ten lettered specimen locations and a ten-step guided tour.

Result

The analytical universe 𝒰 with live counts of admitted and statutory-only parcels, EM-only admissions, the parcels a 15 or 50 km R_em placeholder would misclassify against the computed link, and the parcels a minimum operator would falsely exclude.

Validation

Unit tests for equation (4), the maximum and minimum operators, zone assignment, line-of-sight, road-access, RF-link, and UAS-feasibility admission, knife-edge loss behind a ridge, and the refracted horizon on a flat plain; that equation (1) bounds every parcel Pathway 2 admits; that no two roads cross, touch, or share a stretch without a junction, retrace themselves, or run a piece longer than 40 m, on ridge, coastal, and hills scenes; that no road enters water, on a ridge scene and a coastal one; that every road keeps within its class grade limit on a hills scene and a coastal one; that junction arrival times are shortest paths and the access time along every drawn road equals the exact arrival time; that on a ridge scene and a coastal one no lots overlap or reach into a right-of-way or water, every lot lies in the scene with its front line along its road, every building sits inside its lot on a foundation within its own type's relief limit, principal buildings keep their use's dimensions and setbacks, and the office, warehouse, and farmstead placement rules hold; and agreement between per-parcel tests and the test parcel, which exposes every pathway test with the evidence behind it.

Code / Demo

Open the sandbox

Model notes

Terrain

Seeded gradient-noise heightfields at 125 m in three presets: a ridge-and-valley scene with the facility on the valley floor, rolling hills, and a coastal plain beside open water. Relief governs the viewshed and routes the roads; on the coastal plain the refracted horizon leaves a 1.8 m observer a small share of the analysis disc, while distant rising ground still holds R_view at the radius.

Viewshed

A radial sweep of 2,048 rays from the facility, each keeping the steepest terrain angle seen, with Earth curvature and a refraction coefficient of 0.13, draws the footprint; each parcel is then tested with its own sight line to its ground, and the two can disagree at the drape's edge. Because R_view is a maximum, one distant visible slope sets it, while the footprint and its share of the analysis disc show what terrain actually allows.

Road network

A host town a few kilometres from the facility and up to six towns on low, gentle ground at least 10 km apart are joined by an arterial to the host town, a minimum spanning tree of highways, up to three arterial shortcuts, up to four highway exits toward the scene's edges (on the coast some end at the shore), and rural lanes. Each is routed by A* on a 250 m lattice that measures grade every 25 m, refuses water, and refuses any move steeper than its class allows (5% for highways, 7% for arterials, 10% for streets) wherever a route exists. Where none does, a bottleneck search finds the least steep grade a route through the surrounding area must climb, and the route climbs no steeper than that and as little above its class limit as it can. On the hills and coast presets every road keeps within its limit. On the ridge preset, where no route avoids the ridge, about 6 to 14% of highway length climbs above 5%, at 11 to 22% at the steepest, and on about three seeds in ten a stretch of arterial exceeds 7%, at up to about 12%. Corners are rounded only where the rounded road stays dry and no steeper. Towns, and a larger district around the facility that reaches past the one-mile line, get grids of 110 × 220 m blocks, 110 m blocks downtown, laid on dry land out to an irregular town edge wherever a street's grade stays within about 10%. The network is made planar, so every crossing, every road ending on another, and every stretch two roads share becomes a junction or one shared edge, and every road is resampled every 40 m or less. Each piece is driven at its class design speed (100, 70, and 40 km/h; 70 and 50 km/h in town) over its slope length, slowed uphill by 4% per point of grade above 3% and downhill by 2% per point beyond 5%. Dijkstra gives the arrival time at every junction, and the arrival at any point along a road is the earlier of its two ways in, exact because speed is constant within a piece. A parcel's access time is the earliest arrival anywhere on the road within its access distance. The drawn roads, the service area, R_net, and the test parcel's route are all this one network.

Lots and buildings

Lots are laid along each side of each road: downtown office buildings first, then highway and arterial frontage, then town streets, then rural roads. Houses sit on lots of 15 to 25 m frontage and 30 to 45 m depth behind 6 to 9 m front setbacks, one or two storeys under a gable, most with a garage and some with a garden shed. Main-street mixed-use buildings of two to four storeys are built to the front line with party walls. One to four office buildings of eight to twelve storeys stand within 250 m of the centre of each town with a downtown. Warehouses with an attached office line highway and arterial frontage in the outer quarter of town, on lots no deeper than a block, and on deeper lots up to 500 m beyond the edge, 30 m clear of any house lot. Farmsteads of 1.5 to 6 ha, with a farmhouse, a barn on level ground, machine sheds, and often grain bins, sit along rural roads at least 500 m past the edge of any town, at least 1.6 km apart along each side of a road, and never within 800 m of one another. A lot is rejected if it overlaps another, reaches into any road's right-of-way (40, 30, and 18 m by class), strays more than 3 m from the road it fronts (as on a tight bend), touches water, leaves the scene, or cannot fit its building inside the setbacks; a building is rejected if its footprint would need grading across more relief than its type allows: 1.5 m for a house, farmhouse, or attached garage, 1 m for a shopfront, detached garage, barn, or shed, 0.5 m for a grain bin, 2 m for a warehouse slab, and 3 m for an office building over a daylight basement. Unit tests check each condition, building by building against its own limit, on a ridge scene and a coastal one.

RF link

Path loss from a 3 m parcel antenna to a 10 m facility antenna at 433 MHz to 5.8 GHz: free-space loss, single knife-edge diffraction J(ν) at the path's dominant obstruction under a 4/3 effective Earth radius, and 0.4 dB/km of clutter. The link budget is 125 dB at a MODERATE EM rating and 140 dB at HIGH; a LOW rating carries no EM exposure. The per-parcel test samples every half cell; the drape is a coarser field of 720 rays at 375 m steps, and R_em takes the farther of the drape's edge and the farthest lot whose link closes.

UAS feasibility

Illustrative transit ranges of 10, 40, and 100 km and operating altitudes of 120, 450, and 1,500 m for Groups 1 to 3. A launch must be on land within the straight-line transit range and outside prohibited airspace; a piloted launch also needs a clear line of sight, the same geometric test as VIEW, from a 2 m operator antenna on the parcel to the aircraft holding over the facility. An autonomous mission drops that last test.

What is simplified

The platform-group ranges and altitudes are editable sandbox values, not the PP-10 specification. The RF model omits multipath, antenna patterns, and multiple-edge diffraction. Each parcel is tested at one point, its main building's centre at ground level, and buildings neither block sight lines nor raise targets. The command link uses the viewshed's optical refraction rather than a radio horizon. The drive from the road to the parcel is not timed and junctions add no delay. The 250 m routing lattice cannot resolve switchbacks, so ridge crossings can exceed a class's grade limit and a road that zigzags up a slope turns in a sharp V. The scene is 72 km across, so a road that leaves it within T_net truncates R_net. Lots and buildings are synthetic, and buffers are drawn in a local metric frame rather than EPSG:6488 or EPSG:26918. Nothing in the scene is a pilot result.

Things to try

• Predict each lettered specimen's zone before clicking it, then read the four modality tests on its card.
• Zoom into a town: street-fronting lots with houses and garages, a downtown of shops and office buildings, a warehouse strip on the highway at the edge of town, and farmsteads along the rural roads, each lot in its zone colour.
• Place the 1.02-mile parcel with the shadow pathway off, then on, and read its weight.
• Switch the observer from 10 m to 1.8 m: the gold footprint shrinks while R_view barely moves, because one distant visible slope still sets the maximum.
• Raise the carrier frequency from 433 MHz to 5.8 GHz and watch the blue footprint retreat behind every ridge; then compare the placeholder count.
• Switch the UAS mission from piloted to autonomous: the command-link test drops and the violet footprint fills the transit range over land.
• Shorten T_net and watch the service-area corridors and the green roads retract toward the facility, and whole towns change colour as they fall outside it.
• Set the combining rule to min: the envelope collapses to the intersection, and the magenta parcels are what the maximum rule protects.
• Make the facility a military installation and zoom out until the 100- and 150-mile rings appear on the base plate.