Part III — Mapping the objects onto the three phases

Each phase gets the same four headings: what is random, the object from Part II that describes it, the data it consumes, and what falls out.

III.1 Phase A — Discovery¶

What is random: Locations of sources not in the database
Object: Thinned log-Gaussian Cox process (§II.7 + §II.9)
Data: Fine-imager scenes over searched cells; the infrastructure database; the bottom-up inventory, converted into the GP mean $m_g$ (below)

Discovery counts sources, so it needs a finite number of them. Keep $\lambda$ as the CRM base measure of §II.4 (the same one §III.3 integrates against), and give each source two marks: its on-state rate $s_0 u$ , with Lévy measure $\rho(\mathrm{d}u)$ , and its persistence $\pi$ , with conditional distribution $G(\mathrm{d}\pi \mid u)$ . Count only sources above a size floor $u_{\mathrm{floor}}$ (well below $s_{\min}/s_0$ ). Their count intensity and normalised rate distribution are

\lambda_{\ge}(\mathrm{d}x) = R\,\lambda(\mathrm{d}x), \qquad f(\mathrm{d}u) = \frac{\rho(\mathrm{d}u)\,\mathbf{1}[u \ge u_{\mathrm{floor}}]}{R}, \qquad R = \rho\bigl([u_{\mathrm{floor}}, \infty)\bigr).

(2)

A source at $x$ with marks $(u, \pi)$ , covered by clear scenes $i = 1, \dots, n$ so far, is detected at least once with probability

p_{\mathrm{camp}}(x,u,\pi) = 1 - \mathbb{E}\!\left[\prod_{i=1}^{n} \bigl(1 - Z_i\,p_i(x,u)\bigr)\right],

(3)

where $Z_i \in \{0,1\}$ is the source’s on/off state at scene $i$ (with $\mathbb{E}[Z_i] = \pi$ ), and $p_i(x,u)$ is scene $i$ ’s POD (§II.9.2; it varies with wind, viewing geometry, and retrieval sensitivity). The expectation runs over the joint state model of §III.2, and the answer depends on it:

Independent Bernoulli states (scenes far apart relative to the on/off timescale): $1 - \prod_i (1 - \pi\,p_i)$ , which is $1 - (1 - \pi\,p)^{n}$ for identical scenes.
A state that persists through the whole campaign (a long-lived leak): $\pi\,\bigl[1 - \prod_i (1 - p_i)\bigr]$ , which can be far smaller.

Using the independent formula for a strongly correlated source overstates discovery and understates the missed-emission correction. Its mark average is $\bar p(x) = \iint p_{\mathrm{camp}}(x,u,\pi)\,G(\mathrm{d}\pi \mid u)\,f(\mathrm{d}u)$ .

The central identity

By the (marked) thinning theorem, on a searched cell $A$ , conditional on the intensity $\lambda$ :

\begin{aligned} N_{\mathrm{det}}(A) \mid \lambda &\sim \mathrm{Poisson}\!\left(\int_A \bar p(x)\,\lambda_{\ge}(\mathrm{d}x) + \Lambda_{\mathrm{fp}}(A)\right),\\ N_{\mathrm{miss}}(A) \mid \lambda &\sim \mathrm{Poisson}\!\left(\int_A \bigl(1-\bar p(x)\bigr)\,\lambda_{\ge}(\mathrm{d}x)\right), \qquad \text{independent given } \lambda \end{aligned}

(4)

$\Lambda_{\mathrm{fp}}(A)$ is the expected number of false positives in $A$ : the clutter process of §II.9.3, superposed on the real detections. Omitting it lets false plumes inflate the inferred source density and the undiscovered population. When individual detections matter, carry latent real/clutter labels instead.

Under the LGCP, $\lambda$ is random, and marginally the two counts are dependent through it. That dependence is exactly what lets detections (and empty searched cells) inform the posterior over $\lambda$ , and hence the expected number missed. Write the likelihood conditionally on $\lambda$ and integrate over $\lambda$ ; never factor the marginal.

So the posterior over $\lambda$ updates from both detections and searched-but-empty cells, and the expected number still undiscovered (above the floor) is

\int_A \bigl(1 - \bar p(x)\bigr)\,\lambda_{\ge}(\mathrm{d}x)

(5)

computable, not guessed.

Search strategy falls out. The next cell to image is the one maximising the posterior expected inventory emission that is still undiscovered and that the next image would detect,

s_0\,\mathbb{E}\!\left[\int_A \int_0^\infty\!\!\int_0^1 \pi u\,\bigl(1 - p_{\mathrm{camp}}(x,u,\pi)\bigr)\,\pi\,p_{\mathrm{next}}(x,u)\;G(\mathrm{d}\pi \mid u)\,\rho(\mathrm{d}u)\,\lambda(\mathrm{d}x)\right]

(6)

Here $\pi u$ weights each source by its time-averaged contribution, not its on-state rate, and $\pi\,p_{\mathrm{next}}$ is the chance the next scene catches it on and detects it. Keeping $\pi$ inside one integral preserves the dependence between past misses and the next detection, which share the same $\pi$ ; averaging them over $\pi$ separately would misrank intermittent bright sources against persistent faint ones. The objective weights high posterior density (Cox correlation from neighbours) against what the instrument can actually see.

Running example

Choose the floor so that $\lambda_{\ge}(A_1) \approx 15$ , the count of §II.3. With $\bar p \approx 0.3$ , the prior mean of detections is $0.3 \times 15 = 4.5$ . After one clear fine-imager scene over $A_1$ detecting source 1: $N_{\mathrm{obs}}(A_1) = 1$ against that prior mean of 4.5 detectable, so the posterior $\lambda(A_1)$ drops; the Cox prior propagates milder drops to every cell correlated with $A_1$ . $A_2$ and $A_4$ get no direct likelihood term because they are unsearched, but they still move through the kernel: $A_2$ and $A_3$ , both adjacent to $A_1$ , by similar amounts, and the diagonal $A_4$ by less. Expected undiscovered in $A_1 \approx (1 - \bar p)\,\hat\lambda_{\ge}(A_1)$ with mark-averaged $\bar p \approx 0.3$ .

III.2 Phase B — Monitoring¶

What is random: The on/off state and timing of known sources
Object: Per-site Beta–Bernoulli persistence (§II.4.4), a temporal point process of initiations (§II.10.2), or an on/off switching process (§II.10.4), observed through the thinned clock (§II.9.3, §II.10.6)
Data: The sequence of (overpass, cloud, detected, rate) for each known site

Model choice by physics.

Physical behaviour	Model	Key parameter
scheduled venting	renewal	$h$ peaked at 7 d
leak until repair	two-state (semi-)Markov switching (§II.10.4)	on-state duration = repair time
clustered distinct initiations (upsets, cascades)	Hawkes	$\varphi$ decaying over days
random operations	Beta–Bernoulli	$\pi_k$

The identifiability warning

With $n$ overpasses, $n_{\mathrm{clear}}$ clear, $n_{\mathrm{det}}$ detections at site $k$ :

\frac{n_{\mathrm{det}}}{n_{\mathrm{clear}}} \ \text{ estimates } \ \pi_k \cdot p(x_k, s_k), \quad \text{not } \pi_k

(7)

Use the controlled-release detection curve to divide out $p$ ; otherwise report the product and say so. When the POD varies by scene, do not divide by one curve: fit each clear scene’s detection $y_i$ as $\mathrm{Bernoulli}\bigl(\pi_k\,p_i(x_k, s_k)\bigr)$ , or, under independent states, use the moment estimate $\hat\pi_k = \sum_i y_i / \sum_i p_i$ .

Running example — source 3

10 overpasses, 8 clear, 3 detections at $\approx 400\ \mathrm{kg\,h^{-1}}$ . The detection curve gives $p_i(400\ \mathrm{kg\,h^{-1}}) \approx 0.9$ in every clear scene, so the moment estimate is

\hat\pi_3 \approx \frac{\sum_i y_i}{\sum_i p_i} = \frac{3}{8 \times 0.9} \approx 0.42

(8)

against a truth of 0.3. The discrepancy is sampling noise on 8 scenes, and a $\mathrm{Beta}(1,1)$ prior gives a 90 % interval of roughly $(0.15, 0.7)$ . Renewal structure would tighten this if the blowdowns are regular.

III.3 Phase C — Estimation of rates and total¶

What is random: The on-state rates $s_k$ , the diffuse density $e(x)$ , and their sums
Object: The full measure $\mu = \mu_{\mathrm{pt}} + \mu_{\mathrm{df}}$ with a heavy-tailed CRM prior on $\mu_{\mathrm{pt}}$ (§II.4.3), a log-GP prior on $e$ (§II.6), and the per-plume observation model (§II.8.2, §II.9.3)

Term (ii) is a sum over sources nobody has seen, with the marks $(u, \pi)$ and campaign detection probability $p_{\mathrm{camp}}$ of §III.1. By marked thinning, given $\lambda$ the undetected sources form a Poisson process with intensity $\bigl(1 - p_{\mathrm{camp}}\bigr)\,G(\mathrm{d}\pi \mid u)\,\rho(\mathrm{d}u)\,\lambda(\mathrm{d}x)$ , independent of the detected ones. Campbell’s theorem then gives its first two moments:

\begin{aligned} \mathbb{E}\bigl[\text{(ii)} \mid \lambda\bigr] &= s_0 \int_S \int_0^\infty\!\!\int_0^1 \pi u\,\bigl(1 - p_{\mathrm{camp}}(x,u,\pi)\bigr)\,G(\mathrm{d}\pi \mid u)\,\rho(\mathrm{d}u)\,\lambda(\mathrm{d}x),\\ \mathrm{Var}\bigl[\text{(ii)} \mid \lambda\bigr] &= s_0^2 \int_S \int_0^\infty\!\!\int_0^1 (\pi u)^2\,\bigl(1 - p_{\mathrm{camp}}(x,u,\pi)\bigr)\,G(\mathrm{d}\pi \mid u)\,\rho(\mathrm{d}u)\,\lambda(\mathrm{d}x). \end{aligned}

(10)

The posterior over $\lambda$ adds its own spread on top. Never report (i) + (iii) plus a point value for (ii): that silently drops the Poisson/CRM variance of everything below the detection limit.

Detection depends on both marks. Sources 2 and 4 have $p \approx 0$ and fall entirely into (ii). Source 3 is intermittent but bright: with $p_i \approx 0.9$ in each of 8 clear scenes, $1 - (1 - 0.3 \times 0.9)^{8} \approx 0.92$ , so it is almost surely found, even though its time-averaged $123\ \mathrm{kg\,h^{-1}}$ is near the threshold. Collapsing it to one averaged rate would wrongly place it in (ii).

Each term has its own uncertainty:

(i) Monitored points

(ii) Undiscovered points

(iii) Diffuse

Per-plume rate noise $\eta_k$ (log-normal, tens of percent) $\times$ persistence uncertainty from §III.2.

Running example

The toy basin contains only the four listed sources, so its true (ii) is exactly sources 2 and 4. A real basin also has a tail of smaller sources, and the estimate of (ii) always includes one.

Term	Truth ( $\mathrm{kg\,h^{-1}}$ )	Estimate ( $\mathrm{kg\,h^{-1}}$ )	Note
(i) monitored	$0.9\cdot 120 + 0.3\cdot 410 = 231$	$\approx 108 + 167 = 275$ , $\pm\approx 30\,\%$ per source	source 3 from its §III.2 estimate $0.42 \times 400$ ; source 1 held at its true value for brevity
(ii) undiscovered	$35 + 60 = 95$	posterior of the missed-source sum over all rates, weighted by $1 - p_{\mathrm{camp}}$ (here mostly the tail below $100\ \mathrm{kg\,h^{-1}}$ )	matches 95 only if the prior’s missed mass happens to be right
(iii) diffuse	400	$\approx 400 \pm 60$	30-day mapper stack
total	726	$\approx 675 + \text{(ii)}$	(ii) is set by the prior, not the data; (i) is high by sampling noise on 8 scenes

Attribution. When two fine-imager plumes are near each other, DP/Pitman–Yor (§II.5) decides “one source or two”, with assignment weights built from each candidate’s observation rate $\pi_k\,p(x_k, s_k)$ , not its emission fraction (see the warning in §II.5). Pitman–Yor is the safer choice given the heavy tail.

III.4 Scale summary¶

	point (atom)	area (density)
object	$\mu_{\mathrm{pt}} = \sum_k s_k\delta_{x_k}$	$\mu_{\mathrm{df}} = \int e\,\mathrm{d}x$
rate prior	CRM (Gamma / GGP)	log-GP on $e$
persistence prior	persistence marks $\pi_k$ (Beta–Bernoulli / Beta process)	— (persistent)
seen by	fine imager, 1 scene	coarse mapper, stack
detection	threshold on peak	averaging, $1/\sqrt{n}$
time	sampled: $\pi_k s_k$	averaged: persistent
phase	A, B, C	C
fails when	extent $\gtrsim \Delta x$	extent $\lesssim \Delta x$

III.5 The assumptions carrying the structure¶

#	Assumption	Fails when	Repaired by
1	Independence across disjoint regions (§II.2)	shared geology	Cox process (§II.7)
2	Memorylessness in time (§II.10.2, Poisson)	persistent leaks, schedules, clustered upsets	on/off switching (leaks), renewal (schedules), Hawkes (clustered initiations)
3	Point/area is fixed	always: it is relative to $\Delta x$ (§II.8.1)	model at the fine scale and coarsen, never the reverse
4	Overpasses are a fair sample of source state (§II.8.3)	operations correlate with local overpass time	no fix from data alone