Tier V.D — Total emission estimation

Question: Given a population of detected plumes (and the ones we missed), what is the true total emitted mass over a region and time window?

This is the inventory-grade output of plumax — the number that gets reported into national greenhouse-gas inventories, climate models, and policy dashboards. It also requires the most care, because the satellite catalog you start from is systematically biased: detection thinning means the very things you can’t see (small, frequent leaks) are exactly the things that matter for total mass.

The missing-mass paradox¶

The full Monte Carlo proof is in methane_pod/notebooks/03_missing_mass_paradox. The result, in one sentence:

These two biases pull in opposite directions, but they don’t cancel — averaging the wrong thing over the wrong sample size gives you the wrong total. The corrected estimator has to model the thinning explicitly.

The corrected total-mass estimator¶

Given a TMTPP fit (Tier V.B) with posterior $(\lambda, f, P_d)$:

This is the un-thinned total — what would be emitted regardless of detection. Compare to the naive estimator:

with

$M_\text{naive}$ is biased low because:

• $\mathbb{E}[N_\text{detected}] < \mathbb{E}[N_\text{true}]$ (some events missed).

• $\mathbb{E}[Q \mid \text{detected}] > \mathbb{E}[Q]$ (detected events are systematically bigger — heavy tail of $f$).

The two errors compound rather than cancel: the regional total is undercounted, and the per-event mean is inflated. Inverting the POD model is the only way to recover an unbiased total.

Posterior over total mass¶

With NUTS samples $(\lambda^{(s)}, f^{(s)}, P_d^{(s)})$, the posterior over $M_\text{total}(T)$ is:

Reported as posterior median + 95% credible interval. Both integrals are tractable for the standard intensity / mark choices (closed-form for constant $\lambda$ + lognormal $f$; quadrature otherwise).

Validation strategy¶

• MC ground truth (bias direction). Reproduce the qualitative result of the paradox notebook: simulate a known $(\lambda^{*}, f^{*}, P_d^{*})$, compute $M_\text{true}$ exactly, and check that the corrected estimator recovers $M_\text{true}$ while $M_\text{naive}$ is biased low.

• MC ground truth (calibration). Across 1000 replicates of the previous test, the 95% credible interval on $M_\text{total}$ should contain $M_\text{true}$ ~95% of the time.

• Per-satellite sensitivity. Same population, two different $P_d$ (e.g. GHGSat-floor GHGSat Inc., 2016 vs. TROPOMI-floor Veefkind et al., 2012) → corrected estimator should give the same $M_\text{total}$ posterior. The naive estimator gives wildly different $M_\text{naive}$. This is the test that proves the correction is doing its job.

• Real-data benchmark. Once 07_pod_fitting_mcmc lands with IMEO + Tanager data, compare the corrected total for a well-studied basin (Permian) to published bottom-up inventories (U.S. Environmental Protection Agency, 2024Scarpelli et al., 2020, GHGRP) and to top-down inverse-modelling estimates (Maasakkers et al., 2023Jacob et al., 2022, Sherwin et al.). They will disagree; the question is whether the corrected estimator is closer to the top-down number than the naive one.

Module layout¶

Multi-satellite fusion (Tier V.D extension)¶

For a region observed by $K$ satellites, each with its own POD, the unified detection probability is:

This is the “any satellite saw it” probability. Folds into the TMTPP likelihood as a single replacement of $P_d$ with $P_d^{\cup}$. Adds one strong assumption: detections by different satellites are conditionally independent given the leak size — defensible at the population level, possibly violated for clustered super-emitters.

Open questions¶