Topological Data Analysis in Structural Early Warning Monitoring

Catastrophic Risk Reduction and OPEX Optimization in Critical Infrastructure

1 History and Evolution of Topological Data Analysis

Topological Data Analysis (TDA) originates directly from algebraic topology—a branch of pure mathematics pioneered at the turn of the 20th century by Henri Poincaré and others. For nearly a century, topology studied the invariants of continuous spaces under continuous deformations, employing concepts such as homology groups and Betti numbers to characterize “multidimensional holes” and the connectivity of geometric objects. However, while the classical mathematical apparatus operated on continuous and infinite objects, the advent of digital computing at the end of the 20th century confronted researchers with a challenge of an entirely new nature: analyzing finite, noisy, and discrete point clouds derived from empirical measurements. Foundational work in computational geometry and early computational topology during the 1990s (advanced by researchers such as Patrizio Frosini and Vanessa Robins) laid the theoretical groundwork for treating empirical data as discrete samples drawn from unknown differentiable manifolds.

Figure 1: Topological equivalence: continuous deformation of a torus into a coffee mug¹

The defining breakthrough and the birth of TDA as a distinct applied discipline occurred in the early 2000s with the formalization of persistent homology. Seminal papers by Herbert Edelsbrunner, David Letscher, and Afra Zomorodian (2000–2002) introduced the first efficient computational algorithms, while the landmark publication by Gunnar Carlsson and his Stanford University research group (“Topology and Data”, 2005) established the modern research paradigm. The core of this breakthrough was moving away from searching for a single, rigid geometric scale toward tracking the evolution of topological features across a multiscale filtration. Tools such as persistence diagrams and barcodes enabled an objective, perturbation-resilient separation of transient measurement noise from persistent geometric signatures generated by the underlying physical process.

Figure 2: Persistence diagram generation via Rips complexes²

Figure 3: Example of the transformation of a persistence diagram (left) into a persistence landscape (right)³

Over the past decade, TDA has transitioned from advanced mathematical theory into a mature engineering ecosystem. A key milestone was the development of vectorization methods for persistence diagrams—such as persistence landscapes (Bubenik 2015), Euler characteristic curves (ECC), and persistence images—paving the way for direct integration of topological descriptors into machine learning workflows and classical statistics. Concurrently, the emergence of optimized open-source computational libraries (such as GUDHI, developed by INRIA, and giotto-tda) resolved prior computational performance bottlenecks. Today, TDA represents a leading paradigm in nonlinear time series analysis and phase space reconstruction, with direct, mission-critical applications in structural health monitoring (SHM) and predictive diagnostics for civil infrastructure assets.

2 Executive Summary

2.1 Problem Definition

Contemporary geotechnical monitoring systems face a critical problem with false alarms, which paralyzes decision-making processes and generates unjustified operational expenditures. Traditional, widely used threshold-based methods exhibit a high False Positive Rate (based on the author’s extensive SHM analytical practice), leading to:

• Excessive annual costs of unjustified field inspections and unnecessary inspection intensification
• Alarm fatigue—technical personnel ignoring alerts as a result of fatigue induced by high False Positive Rates
• Inability to detect long-term processes—analytical limitations inherent to traditional methods
• Analytical reactiveness—identification of hazards only after critical thresholds are breached

2.2 Solution: Dual-Track TDA Analysis

This document presents an approach leveraging Topological Data Analysis within a dual-track architecture:

Track A – Sensor Health: Instrumentation diagnostics on raw data

• Continuous assessment of measurement channel integrity and degradation
• Elimination of false alarms caused by equipment malfunctions and transmission disturbances

Track B – Structural Behavior: Analysis of geotechnical processes on calibrated data

• Early detection of seepage anomalies and internal erosion prior to reaching critical phases
• Multidimensional representation of system states in phase space for long-term processes
• Low-frequency spectral analysis of geotechnical signals to detect the Critical Slowing Down effect (spectral power increase in the lowest frequency band)

Cross-Verification Engine: Intelligent decision logic eliminating interpretative ambiguity through cross-verification of outputs from both tracks.

2.3 Key Benefits

Metric	Current State (Threshold-based methods)	Expected Impact (Dual-Track TDA)
False Positive Rate (FPR)	High (est. 70–90%)	Reduction to <10% (target <5%)
Number of unjustified inspections	High frequency during fluctuation periods	Reduction by ~70–80%
Long-term process detection	Reactive (only after threshold breach)	Predictive (months in advance)
Measurement channel availability	~90–95% (unplanned outages)	>99% (predictive instrument maintenance)
Operational efficiency (OPEX)	High reactive maintenance costs	Multiple return on investment (ROI)

Table Explanations:

• False Positive Rate (70–90%) – estimated value based on operating experience with SHM systems in hydrotechnical and geotechnical structures. The exact baseline for a specific facility is determined during the pilot phase via an audit of historical alerts.
• Reduction of unjustified inspections – direct result of eliminating false alarms caused by instrumentation faults, network noise, and diurnal/seasonal cycles; transition from ad-hoc interventions to planned maintenance.
• Long-term process detection – ability to identify slow geodynamic phenomena (e.g., suffosion / internal erosion, consolidation) at early formation stages in phase space, well before signals reach physical alarm thresholds.
• Measurement channel availability – percentage of fully functional and reliable measurement channels. The increase stems from proactive detection of sensor degradation (Track A) and equipment replacement before critical failure occurs.
• Operational efficiency – high return on investment resulting from the cumulative reduction of unwarranted field crew deployments, elimination of unnecessary operational standstills, and protection against the costs of catastrophic failures.

3 Strategic Recommendation

3.1 Phase I: Pilot Implementation

• Scope: A representative group of sensors in a selected critical sector of the facility.
• Method: Parallel monitoring alongside the existing threshold-based system, enabling a direct comparison of sensitivity and stability between both approaches.
• Primary Objective: Validation of TDA algorithms on live industrial data streams—empirical verification of the False Positive Rate reduction and confirmation of lead time in anomaly detection.
• Pilot Deliverables: Rigorous, quantifiable quality metrics serving as an objective decision criterion prior to committing resources to full-scale deployment.

The primary value of the pilot phase lies not in immediate operational savings, but in a drastic reduction of deployment risk. It enables verification of algorithmic assumptions on a micro scale before committing to a facility-wide overhaul of the analytical infrastructure.

3.2 Phase II: Full-Scale Deployment

Following successful indicator validation during the pilot, the system is rolled out across the entire monitoring network of the facility. The primary categories of operational impacts include:

Benefit Category	Impact on Operations and OPEX	Horizon and Verifiability
Field inspection optimization	Drastic reduction of service crew dispatches triggered by false alarms (~70–80% reduction)	Immediate (direct consequence of FPR reduction)
Destructive process prevention	Early detection of seepage phenomena—transitioning from costly emergency repairs to planned remediation	Medium-term (governed by ground dynamics)
Facility operational continuity	Elimination of unnecessary work stoppages or operational standstills caused by erroneous readings	Continuous (stabilization of operating regimes)
Instrument life-cycle extension	Early diagnostics of measurement channel degradation and planned replacement versus critical failure remediation	Continuous (predictive instrument management)

Investment Impact: Full deployment delivers an asymmetrically favorable return profile—the optimization of ongoing operating expenditures (OPEX) offsets deployment capital expenditures over short-to-medium-term horizons.

3.3 Catastrophic Risk Mitigation

A distinct, paramount value proposition of the TDA system is addressing low-probability, high-consequence phenomena with catastrophic structural and environmental impacts (tail-risk events, analogous to the Mount Polley or Brumadinho failures).

A potential breach of an earthen structure entails irreversible ecological damage, loss of operating license, and multi-year legal and financial liabilities. The capability of TDA to provide months of lead time prior to reaching a critical state constitutes a strategic safety policy that transcends standard ROI calculations.

4 Current State and Research Problem

A key challenge in the contemporary structural health monitoring (SHM) paradigm is that dominant analytical methods remain inherently reactive. Inference typically relies on post-hoc historical analysis, causing critical states to be identified with a latency relative to the actual onset of structural instability. Current approaches to geotechnical time-series analysis focus almost exclusively on tracking threshold (amplitude) values and their time derivatives (daily, hourly, or minute rates of change).

Such methodology disregards signal morphology—its intrinsic geometric structure and the dynamics of shape deformation in phase space (a multidimensional representation of system states). Consequently, monitoring engineers lack objective tools to capture the point of bifurcation (a tipping point in system dynamics) where an asset begins drifting toward instability long before breaching physical alarm thresholds.

The inability to distinguish genuine structural changes from environmental noise, diurnal and seasonal cycles, and measurement disturbances produces the operational challenges outlined in the Executive Summary, notably:

• High False Positive Rate (FPR)
• Decision paralysis among technical personnel
• Excessive field inspection costs

These fundamental limitations of incumbent methods provide the primary rationale for applying Topological Data Analysis, which provides a mathematical framework capable of tracking the evolution of shape and multidimensional data structure both in reconstructed phase space and in the native multidimensional measurement space \(\mathbb{R}^d\).

5 Telemetry Data Flow and Proposed TDA Module Integration Architecture

The following diagram illustrates the telemetry data flow architecture in the proposed system—from the on-site I&M instrumentation, through the edge acquisition and transmission layer, to the central server environment (cloud or on-premise).

The integration of the Dual-Track TDA module is designed as an analytical overlay, ensuring a completely non-invasive deployment:

1. Edge layer and acquisition: Signals from vibrating wire sensors and weather/barometric stations are delivered to local telemetry nodes, where initial buffering and transmission take place using wireless protocols (LoRaWAN, cellular, or satellite).
2. Central data repository: The received telemetry stream is archived in the measurement database as raw data and subjected to calibration and thermal-barometric compensation procedures.
3. TDA analytical module integration: The Dual-Track TDA engine consumes data directly from the central database in parallel with classical threshold algorithms:
   • Raw data feeds Track A (Sensor Health) for continuous assessment of transducer and transmission path integrity,
   • Compensated data feeds Track B (Structural Behavior) to track the multidimensional dynamics of the soil-water medium.
4. Decision and presentation layer: The cross-verification results (Cross-Verification Engine) feed the dispatch dashboard and can enrich existing SCADA systems with verified, denoised predictive alerts without requiring hardware infrastructure modifications.

6 Why TDA? A Fundamental Paradigm Shift in Data Analysis

6.1 Limitations of Classical Data Analysis and Statistics

Traditional time series analysis methods and common machine learning algorithms rely on assumptions that become critical bottlenecks in real-world civil engineering infrastructure:

1. The Threshold Trap: Pointwise analysis (e.g., threshold exceedances, simple moving averages, or \(\pm 2\sigma\) deviations) assumes that critical states manifest through sudden amplitude changes. In nonlinear systems, degradation processes (e.g., suffosion / internal erosion, micro-cracking, material fatigue) develop over extended periods without noticeable increases in signal energy—only the internal organization of the system changes.
2. Assumption of Linearity and Simple Correlations: Classical analytical and statistical algorithms assume linear dependencies among measured variables. In complex soil-water media, relationships between pore water pressure, temperature, and wire tension are hysteretic, nonlinear, and strongly coupled, causing simple correlation metrics to fail.
3. The Scale Selection Dilemma: Classical bandpass filters require arbitrary definitions of time windows or noise cutoff thresholds. An overly narrow window generates false alarms triggered by transient fluctuations, while an excessively wide window averages out the signal and masks the early stages of anomalies.

NoteFoundational Premise of TDA: Data Has Shape, and Shape Has Meaning

Topological Data Analysis does not ask what the numerical value is at a given instant, but rather what the geometric shape of the data cloud is in phase space / state space, and how this shape evolves over time.

6.2 The Three Pillars of TDA: What Topology Brings to Data Engineering

TDA introduces three unique mathematical properties to data analytics that no classical statistical method possesses:

6.2.1 1. Coordinate Invariance and Geometric Robustness

Topology investigates system properties that remain invariant under continuous transformations of phase space / state space. In structural health monitoring practice, this implies that:

• Identification of the physical structure of a phenomenon is invariant to constant signal offsets, slow zero-drift, or rotation of the measurement coordinate frame.

• If a hysteresis loop emerges in system dynamics (asymmetric lag in system response during loading-unloading or wetting-drying cycles) or a limit cycle forms, TDA detects the presence of this structure irrespective of local transducer nonlinearities or mutual phase shifts among sensors.

NoteIntuitive Explanation

Traditional algorithms trigger alarms when a signal crosses a designated line on a chart; TDA, by contrast, identifies whether the underlying physical structure of asset behavior is preserved, disregarding the fact that the entire signal may be slowly shifting or drifting due to instrumentation aging.

6.2.2 2. Multiscale Analysis Without Arbitrary Thresholds

Rather than arbitrarily defining a rigid noise threshold or cutoff band, persistent homology analyzes data across all spatial scales simultaneously via a filtration parameter \(\varepsilon\):

• Geometric features that appear and vanish almost immediately across minor increases in \(\varepsilon\) (low persistence \(death - birth\)) constitute mathematically defined measurement noise.

• Features that persist across a wide scale range (high persistence) represent fundamental, enduring physical properties of the system.

Consequently, TDA objectively separates instrumentation interference from true ground dynamics, eliminating the need for manual filter tuning.

6.2.3 3. Detection of Multidimensional Cyclic Structures

While classical statistics and clustering algorithms measure only point compactness or dispersion, TDA identifies multidimensional \(H_1\) loops and complex nonlinear trajectories in \(\mathbb{R}^d\):

• In multisensor arrays, linear correlations capture only pairwise couplings. \(H_1\) loops in multisensor space represent asynchronous, delayed hydrodynamic couplings—the direct signature of seepage bottleneck formation or irregular pore water pressure migration within the dam embankment.

6.3 What TDA Delivers: A Value-Add Matrix

Engineering Challenge	Traditional Methods (Threshold-based / SCADA)	Traditional Machine Learning (ML)	Topological Data Analysis (TDA)
Data dimensionality	Projection to 1D (single-channel analysis)	Dimensionality reduction (PCA, t-SNE) with loss of nonlinearity	Full multidimensional representation of geometry in \(\mathbb{R}^d\) preserving phase information
Noise and interference	Dependent on manually selected filters	Requires extensive data cleaning and pre-filtering	Inherent separation of signal from noise on persistence diagrams (short vs. long persistence)
Early warning (EWS)	None – strictly post-factum detection (after limit breach)	Requires historical labeled failure datasets	Detection of bifurcations and Critical Slowing Down via persistence landscape evolution (unsupervised anomaly detection)
Robustness to sensor drift	False alarms caused by transducer degradation	Erroneous predictions in amplitude-based models	Topological separation: wire noise distortion (\(H_0\)) vs. phenomenon trajectory distortion (\(H_1\))
Interpretability	High (but linear)	Low (black-box neural networks / random forests)	Mathematically rigorous: stability guaranteed by Bottleneck / Wasserstein metrics and \(L^p\) norms

6.4 Bridge to Quantitative Analytics: Topology Vectorization

A historical hurdle to engineering adoption of TDA was the inability to directly feed persistence diagrams into numerical pipelines and decision algorithms. The engineering breakthrough underpinning this system is the vectorization of persistence landscapes:

1. Projection into Banach Space (\(L^p\)): Persistence diagrams are transformed into triangular functions forming landscapes \(\lambda_k(x)\). This enables defining a unique, continuous, and stable norm (a measure of “topological energy”) of the asset state: \[\|\lambda(X_n)\|_p = \left( \sum_{k=1}^\infty \int_{\mathbb{R}} |\lambda_k(x)|^p \, dx \right)^{1/p}\]

TipEngineering Intuition of the Formula

This formula sums the areas beneath all “peaks” of the persistence landscape. In practice, it aggregates the entire multidimensional geometry of the system into a single, well-conditioned scalar representing the cumulative “topological energy” of the cycles and hydrodynamic couplings present in the data.

2. Quantification of Critical Slowing Down and Statistical Trend Detection: By reducing multidimensional geometry to a stable series of norms \(\|\lambda\|_p\), one can directly examine the dynamics of the attractor itself. Increases in power spectral density (PSD) within the lowest frequency band and growing variance in the norm series constitute the empirical signature of Critical Slowing Down (loss of hydrodynamic system resilience).

NoteWhat is an Attractor?

It can be conceptualized as a multidimensional equilibrium pattern of dam hydrodynamics, capturing natural pressure relationships and seepage lag dynamics across different structural zones.

To render the early warning process fully automated and free of arbitrary alarm thresholds, the directionality of these indicator trends is evaluated using statistical testing. This method enables objective assessment of trend monotonicity without distribution assumptions and with robust resilience to measurement noise, delivering substantial lead time before system dynamics shift into a critical state.

7 Physical and Geometric Rationale for Dual-Track Analysis

7.1 Metric Isotropy and Simplicial Complex Filtration for Vibrating Wire Pressure Transducers

A critical challenge in constructing a multidimensional phase space / state space in TDA is defining an appropriate distance metric. Standard Vietoris–Rips filtration relies on the Euclidean distance between points in the cloud, expanding simplicial complexes based on a scalar radius \(\varepsilon\):

ImportantNonlinearity of the Vibrating Wire Transducer

The relationship between the natural frequency of the vibrating wire (\(f\)) and pore water pressure (\(P\)) is quadratic: \[ P \propto f^2 \]

If the phase space / state space were reconstructed directly from the raw frequency variable expressed in \(Hz\), this space would exhibit pronounced anisotropy:

• At low wire tension (low \(f\)), an identical pressure change \(\Delta P\) induces a substantially larger frequency shift \(\Delta f\) than at high tension.
• Consequently, a constant filtration parameter \(\varepsilon\) in raw frequency space would correspond to varying strain energy across different operating points of the transducer.

For this reason, analyzing physical processes requires metric normalization by converting raw frequencies into linear units (Linear Digits \(\propto f^2/1000\)) or, ultimately, fully compensated pore water pressure (\(kPa\)). This ensures metric isotropy—the Euclidean distance in the reconstructed space preserves identical physical meaning across the entire measurement range.

7.2 Track A: Sensor Health – Instrumentation Diagnostics on Raw Data

Analyzing raw signals (excitation frequency, the relationship between tension and resonant oscillation amplitude) focuses directly on the behavior of the mechanical oscillator itself (the vibrating wire and its transmission path):

• Sensor degradation detection: A properly functioning wire generates a regular, stable deterministic attractor in phase space / state space.
• Topological failure signature: Mechanical damage, wire pitting corrosion, tension relaxation, or insulation breakdown along the transmission line introduce asynchronous disturbances and waveform distortions into oscillations. On persistence diagrams, this manifests as a rapid proliferation of short-lived \(H_0\) components (increased noise entropy) accompanied by the degradation of persistent \(H_1\) loops, enabling definitive identification of instrument defects well before complete telemetry loss occurs.

7.3 Track B: Structural Behavior – Geotechnical Medium Diagnostics on Compensated Data

Feeding unified data—corrected pore water pressure (\(kPa\)) following complete thermal and barometric compensation—into TDA algorithms shifts the analytical focus from the device level to physical processes within the soil-water medium:

• Local invariance: A unified pressure metric renders the analytical model invariant to sensor-specific baseline characteristics and differing installation depths.
• Geotechnical attractor topology: The pressure trajectory within the reconstructed phase space / state space reflects the hydrodynamic dynamics of the rock/soil mass. Shifts in persistence landscapes and the evolution of topological norms enable early detection of slow bifurcation phenomena (e.g., onset of suffosion / internal erosion, seepage barrier degradation, and the Critical Slowing Down effect) months before physical alarm thresholds are breached.

8 Concept of Integrating Topological Data Analysis with Geotechnical Monitoring Systems

• Dual-Track TDA Architecture: The system performs analysis across two distinct levels of measurement data abstraction. This separation enables simultaneous, independent tracking of hydrodynamic processes in the rock/soil mass and verification of the physical integrity of the measurement channel itself, eliminating a systemic flaw of traditional threshold-based methods.

• Instrumentation Diagnostics Module (Track A – Sensor Health): Connected directly downstream of the raw data acquisition layer. It monitors the topological invariants of the vibrating wire transducer oscillation signal and transmission noise via simplicial complex filtration. This enables early detection of hardware anomalies—such as wire pitting corrosion, progressive tension relaxation, or lead wire degradation—by tracking step-increases in persistence entropy and the decay of stable limit cycles before telemetry continuity is lost.

• Structural Analysis Module (Track B – Structural Behavior): Operates on physical data following complete calibration and thermal-barometric compensation. Based on phase space / state space reconstruction or vectorization of multisensor point clouds, the module investigates the multidimensional evolution of the system attractor using persistence landscapes. By tracking norms and early warning signatures, including the Critical Slowing Down effect, the module identifies bifurcations and slow destructive processes (suffosion / internal erosion, loss of seepage stability) well in advance of static alarm thresholds.

• Cross-Verification Engine and False Positive Rate Reduction (Cross-Verification Engine): Employing a deterministic, sequential decision matrix (\(A \rightarrow B\)) resolves the issue of alarm fatigue. Pressure anomalies are classified as genuine geotechnical hazards only when the topological stability of the measurement channel is confirmed. Any signal distortions resulting from instrument drift, electromagnetic interference, or environmental cycles are automatically isolated, enabling a drastic reduction in the false alarm rate.

• Technology Stack, Quantification, and Reporting: An analytical pipeline based on optimized computational engines transforms complex persistent homology descriptors into scalable statistical and energy metrics. The results undergo automated synthesis in technical reporting engines, providing objective, auditable asset stability indicators for engineering personnel and geotechnical supervision.

9 Component Architecture with Analytical Track Separation – Dual-Track Analysis

The analytical engine separates measurement data processing into two independent, sequential computational tracks driven by TDA algorithms:

1. Track A (Sensor Health) – physical channel and I&M instrumentation integrity diagnostics on raw data.

2. Track B (Structural Behavior) – analysis of hydrodynamic and deformation phenomena within the soil-water medium on compensated data.

9.1 Logical Data Flow Concept

The following diagram illustrates the routing of the raw telemetry stream into the hardware diagnostic and geotechnical analysis tracks, whose outputs converge in the Cross-Verification Engine:

9.2 Deployment Architecture of Analytical Modules

The analytical module interfaces with the existing data repository infrastructure, executing sequential computational orchestration:

9.3 Decision Logic (Cross-Verification Engine)

The final assessment of monitored asset health derives from a deterministic cross-verification matrix \(A \rightarrow B\). A confirmed operational state and stability of the I&M instrumentation is a necessary condition for qualifying an anomaly as a genuine geotechnical hazard:

Track A Status	Track B Status	System Qualification	Operational Procedure
Channel operational	Stable	✅ Nominal	Routine supervision; maintaining standard operating regime.
Channel fault	Anomaly	🔧 False positive	Elimination of false geotechnical alert. Order channel service or verify telemetry node.
Channel operational	Anomaly	🚨 Geotechnical Alarm	Genuine threat to structural stability. Initiate warning procedures and targeted inspection.
Channel fault	Stable	⚠️ Technical Warning	Scheduled diagnostics of instrumentation and power circuit prior to loss of measurement continuity.

10 Resilient IoT Infrastructure: Coupling the Elixir/BEAM Platform with the TDA Analytics Engine

The effectiveness of advanced early warning algorithms directly depends on the stability and continuity of the data stream feeding them. In large-scale hydrotechnical and industrial assets (such as earthen dams or tailings storage facilities (TSF)), telemetry infrastructure must operate under severe environmental conditions: unstable wireless connectivity, periodic power interruptions, and transmission interference across vast geographical areas.

To address these challenges, the proposed architecture separates system responsibilities into two specialized pillars:

1. Acquisition, orchestration, and telemetry (IoT) layer – powered by the Elixir / BEAM platform,

2. Advanced multidimensional analytics layer – powered by the Python computational ecosystem.

10.1 Telecommunications Heritage: 99.999% Reliability in Structural Monitoring

The Elixir/BEAM ecosystem originates directly from Ericsson’s telecommunications systems, designed to power digital switches requiring near 100% availability (“five nines”). Today, the same technology stack drives mission-critical, large-scale global communication platforms such as WhatsApp and Discord, processing tens of billions of events daily.

In the context of structural health monitoring of hydrotechnical structures, these characteristics translate into distinct operational advantages:

• Virtual isolation of measurement points: Every sensor, base station, and telemetry node (datalogger) is represented within the system as a fully isolated, lightweight virtual microprocess. A failure, read timeout, or physical fault in a single vibrating wire transducer in one dam sector has zero impact on monitoring stability across remaining sectors.
• Self-healing architecture: The system features built-in supervision trees providing automated restart and recovery logic for components experiencing transient faults (e.g., due to power surges, radio transmission timeouts, or battery voltage drops). This eliminates the need for manual on-site maintenance dispatches simply to “power-cycle a hung device”.
• Resource footprint optimization: Edge node software (e.g., running on industrial base stations or single-board computers such as Raspberry Pi) under the BEAM virtual machine operates with minimal CPU and power overhead. This enables persistent local buffering of measurement records during extended backhaul outages (cellular LTE, LoRaWAN, satellite) and guaranteed zero-loss synchronization once connectivity is restored.

10.2 Separation of Concerns: Elixir/BEAM and Python

A foundational architectural tenet is the strict separation of responsibilities: telemetry transport, ingestion, and stream orchestration do not compete for computing resources with intensive mathematical workloads:

11 Technological Positioning and Comparative Analysis

Contemporary geotechnical and hydrotechnical monitoring systems (SHM) for hydrotechnical assets, earthen dams, and tailings storage facilities (TSF) rely on mature telemetry solutions. However, these solutions focus primarily on the data acquisition and transmission layers, leaving the analytical layer at the level of elementary threshold rules.

11.1 Classification of Analytical Paradigms in SHM

Monitoring systems available on the market can be categorized into three technological generations:

1. Telemetry and Datalogger Systems

   • Architecture: Focused on hardware channel reliability under harsh environmental conditions.

   • Analytical methodology: Deterministic alarm thresholds (amplitude-based) and simple time derivatives (\(\Delta P/\Delta t\)).

   • Limitations: Lack of automated mechanisms to distinguish instrument drift from genuine geodynamic phenomena; absence of multisensor coupling modeling.

2. Cloud IoT and Data Aggregation Platforms

   • Architecture: Advanced wireless data transmission (LoRaWAN), cloud-based measurement centralization, and real-time visualization.

   • Analytical methodology: Statistical rules (standard deviations \(\pm k\sigma\)), linear regressions, and cross-correlations of environmental indicators.

   • Limitations: Sensitivity to soil-water medium nonlinearities; high false alarm rate triggered by diurnal/seasonal fluctuations and telemetry noise.

3. Topological Analytics Layer (Dual-Track TDA)

   • Architecture: Independent, overlying algorithmic layer integrated into existing data repositories (vendor-agnostic regarding monitoring hardware).

   • Analytical methodology: Multidimensional phase space / state space reconstruction, persistent homology (\(H_0, H_1\)), and vectorization of persistence landscapes into Banach space (\(L^p\)).

   • Characteristics: Simultaneous assessment of sensor physical integrity (Track A) and seepage-deformation process dynamics (Track B).

11.2 Comparison of Metrological and Analytical Properties

System Feature	Classical Threshold Telemetry	Cloud IoT / SCADA Platforms	Dual-Track TDA Engine
Inference model	Pointwise / One-dimensional (1D)	Spatial / Statistical	Multidimensional state space (\(\mathbb{R}^d\))
False alarm rate	High	Medium / High	Low
Instrumentation self-diagnostics	Binary state detection (no signal / short circuit)	Statistical analysis of transmission continuity	Continuous detection of channel degradation and zero-drift (\(H_0/H_1\))
Anomaly prediction horizon	Reactive	Short-term	Early warning
Nonlinear process detection	Unsupported (masked by averaging)	Partial (sensitive to hysteresis)	Inherent (invariant to continuous deformations)
Infrastructure impact	Requires dedicated hardware	Dependent on vendor ecosystem	Backward compatible (analytical overlay)

11.3 Integration Model with Existing Infrastructure

Implementation of the TDA engine does not require replacing installed I&M instrumentation (vibrating wire piezometers, inclinometers, or extensometers) or modifying telemetry nodes.

The architecture is based on data-layer integration:

• The Cross-Verification Engine concurrently ingests compensated physical values and raw vibration/frequency parameters from the existing database.

• Topological analysis results are returned to the higher-level dispatch system (SCADA / reporting platform) as standardized stability metrics and verified alerts.

This approach enables hydrotechnical asset operators to preserve existing investments in monitoring infrastructure while eliminating the operational expenditures generated by the unreliability of classical threshold-based methods.

12 Scenario Study: Application of TDA to Large-Scale Engineering Assets

NoteMethodological Note

This section presents a conceptual scenario study based on publicly available engineering parameters and the author’s professional experience. The presented assumptions constitute an analytical reference model used to quantify risks and validate pre-deployment assumptions (Phase I).

12.1 Characteristics of the Reference Asset

12.1.1 Exemplary Tailings Storage Facility (TSF) as a Large-Scale System

Tailings storage facilities (TSF) represent one of the most demanding engineering proving grounds for stability and seepage monitoring. The sheer scale of such assets imposes unprecedented requirements on the density of I&M instrumentation.

• Facility surface area: ~2,200 ha
• Maximum dam height: ~80 m
• Volume of deposited tailings: ~1 billion m³
• Piezometer network: multi-level network of vibrating wire piezometers distributed across the crest, dam berms, and native foundation ground
• Environmental exposure: immediate proximity to rivers, agricultural land, and populated areas

12.1.2 Global Context of Catastrophic Risk

Lessons learned across the international mining industry—most notably the catastrophic failures at Mount Polley (Canada, 2014; dam failure triggered by unaccounted foundation shearing) and Brumadinho (Brazil, 2019; rapid liquefaction of an apparently stable structure, 270 fatalities)—demonstrate that:

1. Conventional alarm thresholds fail: Slow structural processes (suffosion / internal erosion, nonlinear consolidation, zonal yielding) evolve without abrupt jumps in pore water pressure until limit equilibrium is breached.
2. Failure costs are fundamentally asymmetric: The financial and environmental impacts of hydrotechnical disasters are measured in hundreds of millions of dollars, dwarfing the lifecycle cost of preventive analytical systems by orders of magnitude.

12.2 Analytical Challenges in the Threshold-Based Regime

12.2.1 Measurement Scale and Alarm Fatigue

In an extensive piezometer network, generating tens of thousands of telemetry records daily creates critical bottlenecks within the classical SCADA/threshold paradigm:

• High False Alarm Rate: Statistical methods based on \(\pm 2\sigma\) deviation bands generate hundreds of spurious anomalies annually. These stem from natural barometric fluctuations, diurnal thermal stresses, or power supply fluctuations at telemetry nodes.
• Erosion of Trust: Recurrent false alarms erode the vigilance of geotechnical teams, leading to ignored alerts or arbitrary threshold increases simply to silence notifications.

12.2.2 Operational Costs of Uncertainty

• Field Inspections: Every threshold breach across a dam perimeter spanning dozens of kilometers requires verification procedures (on-site visual inspection, manual readings from open standpipes, deployment of geotechnical engineers). Annually, this imposes a severe drain on operating budgets.
• Operational Dilemmas: Measurement uncertainty can slow dam raise schedules or force temporary suspensions of tailings deposition in sensitive sectors.

12.3 Dual-Track TDA in the Reference Model

Deploying dual-track analysis on the scale of a major facility such as TSF requires separating time-delay embedding parameters in phase space / state space according to the timescale and nature of the phenomena under investigation.

12.3.1 Track A: Sensor Health – Diagnostics via Topological Signatures

In the aggressive chemical and physical environment of a tailings storage facility, I&M instrumentation is exposed to pitting corrosion, wire tension relaxation, and transmission line insulation degradation.

Transducer health diagnostics are built upon the concept of a topological signature:

• Baseline Signature Generation (\(D_0\)): Following sensor installation and ground stabilization, a baseline reference signature is generated and archived in the Topological Signature Database as a nominal benchmark.
• Long-Term Degradation Detection (\(D_t \leftrightarrow D_0\)): Continuous computation of topological metrics \(k(D_t, D_0)\) tracks slow mechanical drift—decay in the persistence of resonant \(H_1\) loops and proliferation of \(H_0\) noise signal progressive corrosion or micro-cracks weeks before complete failure occurs.
• Transient/Impulsive Fault Detection (\(D_t \leftrightarrow D_{t-1}\)): Step-changes in topological distance between adjacent sliding windows identify sudden telemetry channel faults (e.g., cable insulation puncture, junction box flooding, or power supply surges).

12.3.2 Track B: Structural Behavior – Tracking the Geodynamic Attractor

Monitoring seepage processes within the dam embankment focuses on the multiscale dynamics of the soil-water medium, completely decoupled from hardware interference:

• Phase Space / State Space Representation: Measurement points are analyzed in a multidimensional space \(\mathbb{R}^d\) representing the dam cross-section (where \(d\) denotes the number of correlated piezometers in a given section) or via phase space reconstruction of the attractor. The evolution of the point cloud shape in a sliding time window reflects shifts in the overall hydrodynamic regime.
• Bifurcation Quantification: Computation of a continuous, well-defined topological energy series. Any reorganization of seepage flow networks (e.g., formation of preferential flow paths, drain clogging, or emergence of hysteresis loops during impoundment filling cycles) manifests as a measurable shift in this norm.
• Critical Slowing Down Detection: Power spectral density analysis of the norm series reveals the Critical Slowing Down effect. This provides a direct, threshold-free indicator of decaying hydrodynamic resilience months before any physical breach of seepage stability occurs.

12.3.3 Sensor Agnosticism and Auxiliary Data Sources

The Dual-Track TDA architecture is not restricted to vibrating wire piezometers. The capability of persistent homology to operate over any metric space \(\mathbb{R}^d\) enables direct, nonlinear data fusion from heterogeneous instrumentation deployed across the facility:

1. Subsurface and surface deformation instrumentation (inclinometers, extensometers, GNSS):
   • Track A: Identification of reading artifacts stemming from probe thermal drift, casing shearing/bending, or GNSS multipath interference.
   • Track B: Tracking hydromechanical coupling. Combining pore water pressure vectors with shear strain vectors in phase space / state space reveals slip surface formation and thixotropic behavior prior to slope instability.
2. Drainage and seepage monitoring systems (measuring weirs, electromagnetic flowmeters):
   • Correlating seepage discharge rates with local hydraulic gradients in phase space / state space.
   • Emergence of new \(H_1\) cycles and asymmetries in persistence landscapes enables early differentiation between drain clogging/colmatation (flow reduction accompanied by pressure buildup) and suffosion / internal erosion (soil particle washout, flow increase, and drop in filtration resistance).
3. Seismic and induced seismicity monitoring (accelerometers / seismometers):
   • Tailings storage facilities are often situated in areas of mining-induced seismic activity.
   • TDA enables investigation of rock mass stress relaxation dynamics following seismic events: by comparing the Wasserstein distance between pre- and post-event landscapes, the system verifies whether the asset has returned to its baseline stability attractor or undergone irreversible structural transformation.

Through this unified approach, the Cross-Verification Engine evaluates the integrated geometry of the entire dam monitoring profile rather than isolated measurement points, inherently isolating localized transmission disturbances on individual channels.

12.4 Projected Deployment Impacts in a Large-Scale Context

Deploying TDA technology within a dual-track architecture on the scale of a major facility such as TSF delivers measurable benefits across several key dimensions:

Metric Area	Reference Model Prior to TDA Deployment	Projected Regime Following Full TDA Deployment
False Positive Rate (FPR)	Est. 80–90%	Reduction to <10% via hardware-level filtering (Track A)
I&M Instrumentation Maintenance Strategy	Reactive sensor replacement after communication loss	Predictive servicing based on \(H_0/H_1\) signature degradation
Long-Term Process Detection	Limited to expert post-hoc interpretation	Continuous monitoring of bifurcations in the seepage attractor
Field Inspection Optimization	Mandatory verification of every static threshold breach	70–80% reduction in I&M team field dispatches

13 Conclusions

Integrating Topological Data Analysis within a dual-track architecture (Dual-Track TDA) transforms geotechnical monitoring from a reactive, threshold-based alarm system into a predictive early warning capability. By deploying as a data-layer overlay—without requiring physical intervention in existing I&M instrumentation—the solution pairs ongoing operational expenditure (OPEX) reductions with enhanced asset resilience against catastrophic tail-risk events.

© 2026 Michał Janiga. All rights reserved. The text of this document and the system architecture are the property of the author. Third-party graphic materials are used under Creative Commons licenses or the right of quotation in accordance with source citations.

Footnotes

1. D. Burchardt, The concept of legal space: A topological approach to addressing multiple legalities, 2022, DOI: 10.1017/S2045381722000041, licensed under CC BY 4.0.

2. M. Can Yesilli et al., Topological feature vectors for chatter detection in turning processes, 2022, DOI: 10.1007/s00170-021-08242-5.

3. Gang Ma, Using Topological Data Analysis to Process Time-series Data: A Persistent Homology Way, 2020, DOI: 10.1088/1742-6596/1550/3/032082, licensed under CC BY 3.0.