⚡ electrical engineering

Identifiability-Aware Source Apportionment in City-Scale Advection-Diffusion Systems

This paper introduces the Identifiability-Aware Source Apportionment (IASA) framework, which addresses the challenges of sparse urban sensor data and wind-driven transport by formulating a wind-conditioned lagged inverse problem to robustly estimate source contributions, quantify uncertainty, and provide conservative grouping recommendations for indistinguishable sources based on the mathematical properties of the projected response matrix.

Original authors: Ankit Bhardwaj, Lakshminarayanan Subramanian

Published 2026-08-04

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ankit Bhardwaj, Lakshminarayanan Subramanian

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Great Atmospheric Puzzle

Imagine the air above a city as a giant, invisible soup. This soup is constantly being stirred by the wind, which carries smells, dust, and pollution from factories, cars, and even cooking fires. Scientists want to know exactly who put what into the soup. Is the smog coming from the brick kilns downwind, the busy highway, or the local industries? This is called "source apportionment." It's like trying to figure out who spilled the milk, who dropped the cookie, and who knocked over the vase just by looking at the puddle on the floor.

The tricky part is that the "floor" (our sensors) is very sparse. We don't have a camera on every street corner; we only have a few sensors scattered around the city. Furthermore, the wind is a chaotic chef. It mixes everything together, stretches it out, and changes it over time. Sometimes, the smell of a distant factory looks exactly the same as the smell of a nearby car, once the wind has done its mixing. If you just try to guess the recipe based on the smell, you might get a math answer that fits the data perfectly but is completely wrong about who actually made the mess. This paper tackles that specific headache: how do we know if our guess is actually a guess, or if the data is just too messy to tell the difference?

The "Fingerprint" Detective

The authors, Ankit Bhardwaj and Lakshmi Subramanian, have built a new detective tool called IASA (Identifiability-Aware Source Apportionment). Think of it as a super-smart detective who doesn't just try to solve the puzzle, but first checks if the puzzle is even solvable.

In the old way of doing things, scientists would take their list of suspects (traffic, industry, brick kilns, etc.) and run a math equation to see which mix of suspects best explains the pollution readings. The problem is, these equations are "overconfident." They will happily give you a precise answer—even if the wind and the sensor layout make it impossible to tell the suspects apart. It's like a detective who, when the clues are too vague, just picks a suspect at random and says, "It was definitely him!" with 100% certainty.

The new IASA method changes the game. Instead of just asking, "What fits best?" it asks, "Can we actually tell them apart?"

The Magic of the "Fingerprint"
The paper treats every source of pollution (like a specific factory) as having a unique "fingerprint." This fingerprint isn't just a smell; it's a complex pattern of how that pollution moves through the air, gets delayed by the wind, and hits the sensors over time.

The Wind as a Blender: The wind acts like a blender. If you blend a strawberry and a banana, you get a smoothie. If you blend a strawberry and a raspberry, you might get a smoothie that tastes almost the same. If the wind is blowing in a way that makes the "factory smoothie" and the "traffic smoothie" taste identical, you can't tell them apart.
The Background Noise: The air also has a "background" smell (like the general haze of the city) that isn't from any specific local source. The new method first subtracts this background noise, like cleaning a dirty window, to see the actual fingerprints clearly.

The "Can We Tell?" Test
Once the fingerprints are cleaned up, IASA runs a rigorous test. It looks at the math behind the fingerprints to see if they are distinct enough to be separated.

The "Full Column Rank" Check: In simple terms, this checks if every suspect leaves a unique mark. If two suspects leave marks that are too similar (mathematically "coherent"), the system flags them as "indistinguishable."
The Safety Net: If the system finds that it cannot tell the difference between, say, "Brick Kilns" and "Industries," it doesn't guess. Instead, it says, "We can't separate these two, so we will report them as a single group: 'Industrial Sources'." It refuses to give a false, overly detailed answer.

What They Found
The authors tested this on a platform in New Delhi, using real government data from 32 sensors and wind records. They also ran thousands of computer simulations where they knew the "true" answer to see if their tool could find it.

The Results: In their simulations, when the wind and geometry made sources impossible to tell apart, the old methods (like NNLS, CMB, and PMF) confidently gave wrong answers, sometimes off by a huge margin (up to 1.2 in share error). IASA, however, correctly flagged these cases as "unsolvable" and only reported the groups it could actually distinguish. When it did report an answer, it was incredibly accurate, with an error of less than 0.01.
The Real World Test: On the actual New Delhi data, IASA analyzed four weeks of pollution. It found that in the first three weeks, the pollution was mostly from the general population (cooking, heating, etc.), but in the fourth week, it swung dramatically to be dominated by brick kilns. Crucially, the system confirmed that the sensors and wind patterns were good enough to tell these groups apart, so it reported them separately.

The Takeaway
The big lesson here is that accuracy in prediction does not mean accuracy in identification. You can have a model that predicts the pollution level perfectly but still have no idea where it came from.

IASA introduces a new standard: honesty about uncertainty. It tells policymakers, "Here is the most detailed breakdown the sensors and the wind allow us to make. If we can't tell the difference between two sources, we won't pretend we can." It's a shift from "guessing the answer" to "knowing what we can and cannot know," ensuring that decisions about air quality are based on solid, defensible science rather than confident guesses.

Technical Summary: Identifiability-Aware Source Apportionment in City-Scale Advection-Diffusion Systems

Problem Statement

Urban air quality source apportionment is an inverse problem where the goal is to attribute observed pollutant concentrations to specific source groups (e.g., traffic, industry, brick kilns) using sparse sensor networks. The authors identify a critical geometric limitation: even when a model fits sensor data accurately, the underlying source contributions may not be identifiable.

In sparse sensor networks, distinct source groups can induce nearly identical sensor-time signatures due to wind-driven transport, attenuation, background variation, and noise. A strong distant source and a weaker nearby one may become observationally equivalent after transport and background correction. Consequently, standard receptor models (like Chemical Mass Balance or Positive Matrix Factorization) often report confident attributions even when the data cannot distinguish between sources, leading to potentially misleading policy conclusions. The paper frames this not merely as an uncertainty issue, but as a fundamental lack of identifiability in the sensing system.

Methodology

The authors propose Identifiability-Aware Source Apportionment (IASA), a framework that treats inventory-based apportionment as a wind-conditioned, lagged inverse problem. The core methodology involves:

1. Problem Formulation

Inventory-Based Parameterization: The source field is restricted to a finite set of known or proxy source maps ( $S$ ). Time-varying activity is modeled as a nonnegative combination of pre-declared temporal basis functions ( $\Phi$ ). The unknowns are the source–basis coefficients ( $c$ ).
Wind-Conditioned Lagged Response: The model incorporates a transport operator ( $G^\partial$ ) that maps emissions released at time $t-\ell$ to sensor concentrations at time $t$ , conditioned on wind fields. This creates a "lagged" response matrix ( $H^{lag}_\Phi$ ) where each column represents the sensor-time fingerprint of a unit coefficient.
Background Correction: Observed concentrations include regional background and trends. A low-dimensional background basis ( $Q$ ) is projected out to isolate the source signal. The core object of the inverse problem is the projected lagged response matrix:
$\tilde{H}_\Phi = P^\perp_Q H^{lag}_\Phi$
where $P^\perp_Q$ is the projection onto the orthogonal complement of the background space.

2. Identifiability Diagnostics

IASA shifts the focus from "best fit" to "identifiable resolution." It computes a panel of diagnostics from $\tilde{H}_\Phi$ to determine what can be reliably recovered:

Exact Identifiability: Requires $\text{rank}(\tilde{H}_\Phi) = J$ (full column rank).
Noise-Robustness: Governed by the smallest singular value ( $\sigma_J$ ) and the condition number ( $\kappa$ ). Small $\sigma_J$ implies high sensitivity to noise.
Coefficient Visibility: The magnitude of the projected fingerprint ( $v_j = \|\tilde{h}_j\|_2$ ). Coefficients below a threshold are flagged as "weak."
Pairwise Coherence: Measures how proportional two fingerprints are ( $\rho_{ij}$ ). High coherence indicates sources are indistinguishable.
Background Absorption: The fraction of a source's signal lost to the background space ( $a_j$ ).

3. The IASA Algorithm

The procedure enforces a strict separation between fitting and diagnosis:

Pre-declaration: All model choices (background basis, lag window, temporal bases, zero-masks) are fixed before inspecting data.
Fitting: Nonnegative regularized least squares (NNLS) is solved for coefficients $c$ .
Diagnosis: Diagnostics are computed on $\tilde{H}_\Phi$ .
Reporting:
- If coefficients are weak or highly coherent, they are not reported individually.
- A source ambiguity graph is constructed where edges connect indistinguishable pairs.
- Connected components of this graph form "conservative report groups."
- The final output is the finest attribution resolution supported by the sensors, wind, and noise, accompanied by uncertainty intervals and ambiguity certificates.

Key Contributions

Formulation: The paper formalizes sparse-sensor source apportionment as a lagged, wind-conditioned inverse problem, identifying the projected response matrix $\tilde{H}_\Phi$ as the central determinant of separability.
Reporting Protocol: It introduces a concrete protocol that distinguishes predictive accuracy from identifiability. It reports only those contributions that are defensible under the available data, flagging indistinguishable sources and merging them into conservative groups rather than reporting false precision.
Algorithm (IASA): A procedure that fits nonnegative coefficients, reconstructs activity trajectories, quantifies uncertainty, and diagnoses indistinguishable sources. It includes per-sensor spatial footprints that trace transport responses back to upwind regions.
Empirical Validation: The framework is instantiated on a New Delhi platform using government PM2.5 and wind records. It demonstrates that standard baselines (NNLS, CMB, PMF) fail to detect non-identifiable cases, whereas IASA flags every non-identifiable scenario and recovers apportionment within 0.01 share error in controlled settings.

Results

The authors evaluate IASA on a New Delhi platform (32 sensors, 2018–2020 data) and controlled synthetic experiments:

Controlled Experiments:
- Wind/Geometry Collapse: When wind direction is constant or sensor geometry causes source fingerprints to align, standard methods report confident but incorrect splits (error up to 1.2). IASA flags these as non-identifiable and reports the merged group with near-zero error.
- Background Stress: When a background basis absorbs source signals, standard methods misattribute. IASA detects the absorption and flags the loss of identifiability.
- Conditioning: Recovery error is shown to be bounded by the conditioning of $\tilde{H}_\Phi$ (singular values), not just observation noise.
- Lag Selection: The optimal lag window is selected based on the geometry of the response matrix (Frobenius norm stability), not fitted coefficients, ensuring the lag does not artificially improve identifiability.
Observed New Delhi Data:
- Applied to four consecutive one-week windows, the system identified that all four source groups (brick kilns, industry, population, traffic) were individually identifiable in these specific weeks (full rank, low coherence).
- The apportionment results showed significant temporal variability (e.g., population dominance in early weeks, brick kiln dominance in the final week), which was reported separately rather than aggregated, reflecting the actual data support.
- The system provided uncertainty intervals and confirmed that the reported resolution was the finest supported by the declared inventories, transport, and noise.

Significance and Claims

The paper claims that prediction accuracy is not evidence of identifiability. A model can fit sensor trajectories perfectly while attributing pollution to the wrong sources if the sources are geometrically indistinguishable in the sensor space.

The significance of IASA lies in its conservative resolution: it reports the attribution resolution that the sensing system actually supports, conditional on the declared wind, transport, background, and inventory models. It does not claim to solve the inverse problem in all cases; rather, it provides a diagnostic framework to determine when the problem is solvable and what can be reliably reported.

The authors emphasize that their work complements existing receptor and source-oriented models by adding a layer of identifiability analysis. They do not claim to invent new source categories or invent new physical transport models, but rather to rigorously analyze the distinguishability of declared source groups under sparse sensing. The framework ensures that policy decisions are based on defensible, identifiable contributions rather than artifacts of model overfitting or geometric ambiguity.