The most common question we get from procurement teams evaluating a seismic-based underground detection system is some variant of: how many nodes do I need? It sounds like a logistics question but it's actually a physics and probability question. The answer depends on geology, corridor geometry, threat signature characteristics, and the detection probability threshold you're willing to accept. This article walks through how we think about that geometry — including the variables that dominate the answer and the ones that matter less than intuition suggests.
We're framing this as "our modeling suggests" throughout, because the numbers presented here come from internal simulation and controlled field trials, not independently verified deployments at scale. Real-world geology is heterogeneous in ways that any model simplifies. These figures should be treated as planning estimates, not procurement specifications.
The Seismic Sampling Analogy
There is a useful analogy to Nyquist-Shannon sampling theory here. In signal processing, the Nyquist criterion states that you must sample a signal at a rate at least twice the highest frequency of interest to avoid aliasing — to avoid missing signal content that exists between sample points. The spatial equivalent applies to seismic sensor arrays: to reliably detect a seismic source at any point in a corridor, you need sensor spacing no greater than half the maximum detectable source separation distance.
The detectable source separation distance depends on the attenuation properties of the medium. Seismic energy from a walking human — a footfall coupled into the floor of a concrete tunnel — attenuates differently in concrete, soft soil, and competent rock. The faster the signal attenuates, the closer your nodes need to be. The inverse relationship between material stiffness and attenuation rate means that counterintuitively, harder geology often allows wider node spacing because energy travels farther before falling below detection threshold.
The practical complication is that tunnel corridors are not homogeneous waveguides. A concrete-lined tunnel has the lining as one propagation medium, the substrate geology as another, and the air column as a third. Energy couples between these in geometry-dependent ways. Our modeling treats the dominant propagation path as the floor-contact medium, which is where footfall energy concentrates most reliably.
Spacing by Geology Type
Based on our internal field testing and propagation modeling, here are the node spacing regimes we use as planning baselines for different substrate conditions. These are one-dimensional corridor spacings — single-axis arrays along the tunnel length. Coverage width (across the corridor) is a separate geometry handled by sensor placement height and sensitivity directivity.
Reinforced Concrete Lined Corridors: 2–3 m Spacing
Concrete is a low-attenuation medium for seismic signals in the relevant frequency range (roughly 10–200 Hz for walking signatures). Energy propagates efficiently along the slab, and coupling from footfall to slab is direct. Our modeling suggests that in a standard 2.4 m wide military-grade concrete tunnel, a 2 m node spacing achieves detection probability above 90% for a single pedestrian with normal walking mechanics at speeds up to 2 m/s. At 3 m spacing, detection probability in our models drops to approximately 75–80% for the same scenario. We have not validated these numbers against independent third-party field trials.
The 2 m spacing threshold also relates to localization resolution. For triangulation-based position estimation using time-difference of arrival (TDOA) between adjacent nodes, 2 m spacing gives adequate baseline length for meter-scale localization accuracy in concrete — important if you're using seismic data to cue a directed effector response rather than just generating an alert.
Soft Soil and Earthen Corridors: 4–5 m Spacing
Soft, unconsolidated soil — the substrate typical of hand-dug narco tunnels or improvised military corridors in alluvial terrain — attenuates seismic energy significantly faster than concrete. The frequency content of footfall signatures in soft soil also shifts: high-frequency components (above ~80 Hz) are absorbed within 2–3 m, leaving primarily low-frequency (15–50 Hz) energy available at greater distances.
The geophone sensitivity and filtering configuration must be adjusted for this regime. In concrete, you can get reliable results with standard 4.5 Hz geophones. In soft soil, lower natural-frequency sensors (2 Hz range) and appropriate high-pass filtering improve detection probability at spacing above 3 m. Our modeling suggests that 5 m node spacing in typical soft alluvial soil achieves detection probability in the 70–80% range for pedestrian threats at walking pace. At 8 m, coverage drops below 60% and confidence intervals widen substantially — not a deployable configuration for corridors where we expect human-speed threats.
Competent Hard Rock: 7–10 m Spacing
Hard, competent rock — granite, basalt, dense limestone — has very low seismic attenuation for the frequencies of interest. Energy propagates efficiently and coherently over longer distances. In practice, the detection range ceiling in this regime is often set not by attenuation but by ambient noise characteristics (machinery vibration, water flow, micro-seismic activity) rather than signal propagation distance.
In controlled rock tunnel environments where ambient noise is manageable, our modeling suggests that 8–10 m node spacing is achievable while maintaining detection probability above 70% for pedestrian-class threats. Hard rock also gives better TDOA localization geometry per node-pair because the propagation velocity is higher and more predictable, which helps with source position estimation even at wider spacing. The caveat is that this regime is also where false positive rates from geological noise sources are highest, which we address separately in our multi-modal fusion architecture.
The Detection Probability vs. Spacing Curve
Across all geology types, the relationship between node spacing and detection probability follows a similar qualitative shape: nearly flat performance improvement from very close spacing down to the optimal range, then a steep cliff as spacing exceeds the medium's effective propagation distance. The cliff location shifts significantly by geology — roughly 3 m in soft soil, 5 m in concrete, 12+ m in hard rock by our modeling.
This curve shape has a practical implication: there is diminishing return from going below the optimal spacing. Halving spacing from 2 m to 1 m in a concrete corridor does not double detection probability — it increases it marginally while doubling hardware cost and deployment time. The cost-optimal operating point sits near the beginning of the cliff, not far below it.
We're not claiming our curve models are exactly right for every deployment scenario. Heterogeneous geology, partial concrete lining, backfilled sections, and irregular floor surfaces all introduce deviations. The curves are planning tools, not performance guarantees.
Variables That Matter Less Than Expected
Corridor width, up to about 4 m, has less effect on single-axis array spacing than intuition suggests. The dominant propagation is through the floor contact medium; the walls add reflected energy that can actually improve detection at the array by providing additional coupling paths. Width becomes important for cross-corridor localization and for wide corridors (>4 m) where a single-axis array may create a geometric dead zone at the center. For standard 1.5–3 m military and trafficking tunnel dimensions, a single floor-mounted array is usually adequate.
Threat velocity within the 0.5–2.5 m/s range also does not substantially affect the spacing calculation. The relevant question for spacing is not how fast the target moves but whether a sufficient number of footfalls occur within the detection zone of a given node to build signal confidence above threshold. At 5 m spacing and 1 m/s walking speed, a target traverses the inter-node gap in 5 seconds — enough time for 3–5 footfall events. The dwell time per detection zone is adequate for accumulating statistical confidence. Where velocity does matter is in alert latency requirements: a faster-moving threat spends less time in each node's coverage zone, requiring faster signal processing and lower confidence thresholds, which in turn affects false positive rate. That tradeoff is the subject of a separate discussion.
Multi-Node Fusion and Coverage Continuity
Single-node detection architectures treat each node as an independent alarm. That approach is fragile — any single node failure creates a coverage gap. Our architecture fuses signal data across adjacent nodes using a sliding window correlation approach. A detection event requires correlated energy arrival at a minimum of two adjacent nodes within the propagation-speed-consistent time window. This eliminates the vast majority of false positives from localized noise events while maintaining sensitivity to true threats, because real footfall energy radiates outward and appears at adjacent nodes with predictable time delay.
The consequence for spacing is that the effective detection baseline is not a single node's range but the overlap between adjacent node coverage zones. At our recommended spacings, adjacent node coverage zones overlap by roughly 30–40% of the inter-node gap. That overlap is the design target — it provides redundancy against individual node sensitivity variation and ensures the TDOA localization baseline captures at least two nodes reliably.
Deployments where operators have tried to stretch node spacing beyond our recommended ranges to reduce hardware cost have consistently shown performance degradation at the coverage boundary between nodes — not in the center of each node's zone but at the midpoints. That midpoint gap vulnerability is the failure mode we design against with the 30–40% overlap requirement.