Tier V.A — Instantaneous emission estimation

Question: Given one satellite overpass with a detected plume, what is the posterior over the source’s instantaneous emission rate $Q$?

This sub-page is the glue between the per-event physics tiers (I–IV) and the population tier (V). It defines the formal interface that turns a per-event posterior into evidence for the population-scale fit. The cross-tier payload schema is pinned in the Tier V index; this page formalises how each payload is converted into a mark-likelihood contribution.

Two regimes of “instantaneous Q”¶

A. Full-physics posterior (Tier I / II / III output)¶

When you have radiance / column data and the wind field, run the per-event inversion described in Tier I (or II / III for richer transport):

This is the gold-standard per-event evidence — calibrated, with proper UQ, accounting for wind, transport, and instrument noise. Both $L_i(Q)$ and the prior $\pi_\text{per-event}(Q)$ are exposed downstream because the population fit needs to divide out the prior (see § Mark likelihood below).

B. Catalog $Q$ with wind-source consistency rescaling¶

For published plume catalogs (IMEO, Carbon Mapper alerts Carbon Mapper, 2024, Tanager monthly reports, GHGSat releases GHGSat Inc., 2016Jervis et al., 2021), what’s available is typically a single point estimate of $Q$ (mass flux, t/h or kg/s) per detection plus a per-event uncertainty estimate.

Catalog $Q$ is not $Q/U$ — it’s already wind-multiplied. Operational catalogs use the IME (Integrated Mass Enhancement) method Varon et al., 2018: $Q_\text{catalog} \approx \text{IME} \cdot U_\text{catalog} / L_\text{plume}$, where $U_\text{catalog}$ is the wind value the catalog producer used at retrieval time. When fusing catalogs that used different wind sources (IMEO uses GEOS-FP, Tanager uses HRRR, Carbon Mapper uses ECMWF, GHGSat uses its 1 km downscaled product), inconsistent winds give inconsistent $Q$. The fix is wind-source consistency rescaling:

with $U_\text{target}$ from a single agreed-upon reanalysis (default: ERA5).

Per-event uncertainty comes from controlled-release calibration. Sherwin et al. (2024) report 1σ log-scale errors per instrument:

Mark likelihood — importance-weighted Monte Carlo¶

The TMTPP mark-likelihood contribution at detection $i$ is (see Tier V index § TMTPP foundations):

with $L_i(Q) = p(\text{observation}_i \mid Q)$ the per-event likelihood, not the posterior. In sample-based practice (per-event posterior samples $Q_i^{(s)} \sim p(Q \mid \text{observation}_i)$):

The ratio $f / \pi_\text{per-event}$ is the importance weight that re-points the per-event posterior at the population mark distribution.

Three implementation regimes — with importance correction in each¶

Each row evaluates the same $P_d \cdot f / \pi$ integrand; the only difference is the per-event posterior representation.

Regime selection rule¶

def pick_regime(per_event: PosteriorPayload, mark_class: type[MarkDistribution]) -> Regime:
    cv = per_event.coefficient_of_variation()
    if cv < 0.20 and mark_class.is_smooth_on_scale(cv):
        return "point"
    elif cv < 0.50 and mark_class in CONJUGATE_FAMILIES:
        return "gaussian"
    else:
        return "full"      # CV > 0.50 (detection-floor events), multimodal posteriors

Operational rule:

• Point acceptable only when per-event $\operatorname{CV} < 20\%$ AND $f$ is approximately constant on that scale.
• Gaussian summary suffices when $\operatorname{CV} < 50\%$ AND $f \in \{\text{power-law}, \text{lognormal}, \text{gamma}\}$.
• Full posterior required when $\operatorname{CV} > 50\%$ (typical for detection-floor, very-low-$Q$ events), when $f$ has structure on the per-event uncertainty scale, or when the per-event posterior is multimodal.

Detection-floor and non-detection — explicit handling¶

Detected events with posterior mass below the threshold¶

The importance-weighted MC handles this automatically — $P_d(Q^{(s)})$ is small for small $Q^{(s)}$ so the contribution is naturally downweighted. No special code path. Just don’t point-summarise these events; promote to Full-posterior regime.

Non-detection events (catalog gaps)¶

Non-detections do not flow through the per-event payload — there’s no posterior to ingest. They contribute through the integrated thinned-rate term in the TMTPP likelihood:

What the cross-tier interface needs from the catalog: per-instrument overpass coverage (which times each instrument was looking at the basin), so the integral can be computed correctly.

The catalog ingestion module owns the distinction between “non-detect” and “noisy-detect-near-floor” — it’s a catalog/ingestion concern, not a per-event-payload concern.

Multi-source per overpass¶

Tier I Step 6 (RJMCMC), Tier IV §1 ($K = n_\text{sources}$ first-class), and Tier V index all assume $K > 1$ is supported. The mark-likelihood contribution at a multi-source overpass is a product over sources, assuming within-overpass independence:

Per-event payload for $K_i > 1$ is a list of $K_i$ sub-payloads, each with its own samples / summary / prior. v2 relaxes within-overpass independence — sources sharing a met realisation have correlated marks.

Per-event payload schema (full)¶

Pinned in the Tier V index; reproduced here for the implementation cycle:

@dataclass
class PerEventPayload:
    sources: list[SourcePayload]               # K_i entries; K_i ≥ 1
    instrument_id: str                         # per-satellite POD dispatch
    t_detection: float                         # UTC seconds, for λ(t)
    quality: dict                              # bitmask from Tier I–IV

@dataclass
class SourcePayload:
    posterior_samples: jax.Array | None        # (S,) draws of Q
    posterior_summary: tuple[float, float] | None   # (μ_logQ, σ_logQ) lognormal shorthand
    per_event_prior_logpdf: Callable[[float], float]  # required for importance correction
    x0_posterior: tuple[jax.Array, jax.Array]  # (mu_xy, Cov_xy) — for spatial Cox v2 + de-dup

per_event_prior_logpdf is the load-bearing field — without it the importance correction can’t be done.

Catalog ingestion — heterogeneous sources¶

Per-source ingestion adapters because schemas, units, wind sources, and quality conventions differ:

Each ingestion adapter normalises to the internal PerEventPayload, applies wind-source consistency rescaling against $U_\text{ERA5}$, and emits a unified catalog with explicit provenance fields.

De-duplication¶

Same physical leak detected by multiple satellites → multiple catalog rows. v1’s independence assumption (see Tier V index § Independence assumption) requires de-duplication before the population fit. Default rule: spatial-temporal clustering with thresholds $(\Delta d \leq 5\text{ km}, \Delta t \leq 12\text{ h})$. Multi-instrument detections of the same cluster either collapse to one event with a fused per-event payload (preferred, when posteriors are compatible) or to the highest-confidence detection (fallback).

Module layout¶

Validation strategy¶

• Round trip on synthetic releases. Generate a known $Q^{*}$, run Tier I forward → noisy observation → Tier I inversion → check $Q^{*}$ sits in the reported credible region. Standard sanity check.
• Importance-correction round trip. Generate per-event posteriors using one prior $\pi_\text{per-event} = \operatorname{LogNormal}(0, 1.5)$; run the population fit. Re-run with $\pi_\text{per-event} = \operatorname{LogNormal}(0, 0.5)$ (informative). The recovered population posterior on $f$ should not move beyond IS noise. Failure here means the importance correction is mis-implemented — single most diagnostic test.
• Importance-weight ESS diagnostic. Synthetic scenario where $f$ is a wide power-law and per-event posteriors are tight lognormals far from the bulk of $f$. The ESS-per-event report should warn ($\operatorname{ESS} \ll S$); the population fit should not silently absorb biased estimates.
• Regime-selector consistency. All three regimes (Point, Gaussian, Full) on the same per-event posterior should give the same population posterior modulo regime-appropriate noise. Catches importance-correction bugs in the Gaussian closed-form.
• Wind-source consistency rescaling. Synthesize a catalog with $U_\text{catalog} = 1.5 \cdot U_\text{target}$ for half the rows; before rescaling, the population fit on $f$ should be biased; after rescaling, bias gone.
• De-duplication test. Two synthetic catalogs that are identical up to instrument label; correct de-duplication should collapse them to a single-instrument result.
• Catalog-vs-full-inversion bias. Run the full Tier I inversion on a synthetic radiance, compute the $Q$ you’d get from the IME method (Varon et al., 2018) on the same data. Quantify systematic bias direction (Sherwin et al. 2024 documents this; replicate).
• Proxy idempotency. Apply the per-event → mark-likelihood adapter on a degenerate case (point posterior, identity POD, uniform $f$): the population fit should reduce exactly to a uniform-weight max-likelihood fit on the point estimates. Catches indexing / weighting bugs.

Open questions¶