Part II — Building the mathematical objects

Each section introduces exactly one new idea and names the phase it serves.

II.1 Regions and measures — and the point/area split (all phases)¶

Region

$A \subset S$ is any subset you could draw on a map.

Pairwise disjoint

$A_1,\dots,A_n$ with $A_i \cap A_j = \varnothing$ for $i \neq j$. Adjacent cells: disjoint. A cell and the basin containing it: not.

Partition

Pairwise-disjoint regions whose union is $S$. The $2 \times 2$ grid of §I.5 is one.

Measure

$m$ assigns to each region $A$ a number $m(A) \ge 0$ with $m(\varnothing) = 0$ and additivity over pairwise-disjoint regions:

$$m(A_1 \cup A_2 \cup \cdots) = m(A_1) + m(A_2) + \cdots$$

(1)

Measures on $S$: area $\lvert A\rvert$; emission rate $\mu(A)$; source count $N(A)$. Not measures: max rate in $A$; mean $\mathrm{XCH_4}$ over $A$.

Point mass

$\delta_x(A) = 1$ if $x \in A$, else 0.

Why the split matters for modelling. Atoms are described by a point process (§II.3–§II.5). A density $e(x)$ is a field (§II.6). They are different random objects and get different priors. Everything else in this section is organised around that fact.

II.2 Randomising the inventory (all phases)¶

Random measure. Before looking at data, the number, locations, rates, and density are unknown. A random measure is a random variable whose value is an entire measure; for each fixed $A$, $\mu(A)$ is an ordinary positive random variable.

Complete randomness — the one structural assumption. $\mu$ is completely random if for any pairwise-disjoint $A_1,\dots,A_n$, the totals $\mu(A_1),\dots,\mu(A_n)$ are mutually independent. Learning $A_1$ tells you nothing about $A_2$.

II.3 Counting point sources — the Poisson process (Phase A)¶

Ignore rates. $N(A)$ is a Poisson process with intensity measure $\lambda$ if

1. $N(A) \sim \mathrm{Poisson}(\lambda(A))$, where $\lambda(A)$ is the expected number of sources in $A$;

2. counts in pairwise-disjoint regions are independent.

With density: $\lambda(A) = \int_A \lambda(x)\,\mathrm{d}x$, $\lambda(x)$ in $\mathrm{km^{-2}}$. Homogeneous: $\lambda$ constant. Inhomogeneous: high over active fields, $\approx 0$ over rangeland.

$N$ is itself a completely random measure: the case where every atom weighs 1.

II.4 Attaching rates — the completely random measure (Phase C)¶

Work with dimensionless marks $u_k = s_k/s_0$, so that the physical rate is $s_k = s_0 u_k$. Treat the pairs $(x_k, u_k)$ as a Poisson process on $S \times (0,\infty)$ with intensity

Reading. $\nu(A \times [a,b])$ = expected number of sources in $A$ with rate in $[a s_0, b s_0]$. $\lambda$ handles where, $\rho$ handles how big.

What $\rho$ is. Not a probability density. $\rho(\mathrm{d}u)$ is the expected number of sources, per unit of $\lambda$, with dimensionless rate in $\mathrm{d}u$. It may have infinite mass near 0; what is required is

Infinitely many negligible sources, finite total. This is the nonparametric part: the number of sources is never fixed.

The atomic random measure is $\mu_{\mathrm{pt}}(A) = \sum_{k:\,x_k\in A} s_k = s_0 \sum_{k:\,x_k\in A} u_k$, and its law in every region follows from $\rho$ via the Laplace functional (for dimensionless $\theta \ge 0$)

The choice of Lévy measure $\rho$ selects the member of the family:

II.4.1 Poisson process¶

$\rho = \delta_1$. Every atom weighs 1; $\mu_{\mathrm{pt}}(A)/s_0 = N(A)$. A prior over counts.

II.4.2 Gamma process — inventory prior with light tails¶

Take a prior inventory $H$ ($\mathrm{kg\,h^{-1}}$ per region, e.g. from a bottom-up product) and set $\lambda(\mathrm{d}x) = H(\mathrm{d}x)/s_0$ and $\rho(\mathrm{d}u) = u^{-1}e^{-u}\,\mathrm{d}u$. Then

II.4.3 Generalised Gamma — inventory prior with super-emitters¶

Power-law body, exponential cut-off at scale $1/\tau$.

II.4.4 Beta process — persistence prior (Phase B)¶

Every atom is now a probability $\pi_k \in (0,1)$: the fraction of time source $k$ is on. For overpass $i$ draw $Z_{ik} \sim \mathrm{Bernoulli}(\pi_k)$:

The overpass $\times$ source matrix $Z$ (filled = on, hollow = off) is the feature allocation; $c$ controls how many sources are persistent versus flickering. Integrating out the $\pi_k$ gives the IBP on $Z$.^[2]

Known candidate sites — the finite version. When the infrastructure database supplies candidate coordinates $c_1,\dots,c_J$, there is nothing nonparametric about those: model each with $Z_j \sim \mathrm{Bernoulli}(\pi_j)$, $\pi_j \sim \mathrm{Beta}(a,b)$. Sources not in the database (unknown-unknowns) need one marked process: locations from the Poisson/Cox process (§II.3, §II.7), with each atom carrying its own persistence mark $\pi_k \sim G(\mathrm{d}\pi \mid u_k)$. This is the construction used in §III.1.

II.5 Fractions and attribution — normalisation (Phase C)¶

If $0 < \mu_{\mathrm{pt}}(S) < \infty$, then $G(A) = \mu_{\mathrm{pt}}(A)/\mu_{\mathrm{pt}}(S)$ is the fraction of the basin’s point total from $A$. For the Gamma process, ratios of independent Gammas are Dirichlet, so for any partition:

This is the Dirichlet process $\mathrm{DP}(H/s_0)$: its base measure is the dimensionless $H/s_0$, whose total mass $H(S)/s_0$ is the concentration.

Attribution reading. Each detected plume is assigned to a source.

Attribution matters when two plumes in one fine-imager scene could come from one source or two.

II.6 Fields — for concentrations and for area sources (Phases A, C)¶

Three quantities in this problem are fields, not measures:

None of them is additive over regions. All three need the same construction.

Random field. Specify, for every finite set of locations $x_1,\dots,x_n$, the joint law of $(f(x_1),\dots,f(x_n))$, subject to consistency: integrating the $(n+1)$-point law over one coordinate must reproduce the $n$-point law. A consistent family determines a unique stochastic process in distribution.^[3] Consistency is what lets you say “the field”.

Pathwise quantities such as $\int_A e(x)\,\mathrm{d}x$ need more than the law: they need a measurable, integrable modification of the process. For the kernels used here (exponential, Matérn, squared-exponential), the Kolmogorov continuity criterion supplies a continuous one, which is what makes those integrals well defined.

Gaussian process $\mathcal{GP}(m,k)$.

with $k$ positive semi-definite. Consistency is automatic.

Beyond Gaussian.

Student-t process

Multivariate-$t$ finite-dimensional laws with common $\nu_t$; consistent; robust to outlier retrievals. $\nu_t \to \infty$ gives the GP.

Wishart process

$\Sigma(x) = A(x)A(x)^{\mathsf T}$ with GP columns; a spatially varying covariance across jointly retrieved species.

II.7 Composing field and point process — the Cox process (Phase A)¶

Repair §II.2’s independence assumption: let the source density be a random field,

Given $\lambda$, disjoint cells are independent. Marginally they are correlated, because the same $g$ shaped both:

positive wherever $k_g > 0$, as it is everywhere for the usual exponential, Matérn, and squared-exponential kernels, and negative where a kernel with negative lobes has $k_g < 0$. This is the LGCP, and it is the natural discovery prior: more detections in $A_1$ than its search effort predicts raise the posterior source density in adjacent $A_3$ before $A_3$ has been searched, and fewer lower it. The update depends on the count relative to its expectation, because the Poisson likelihood pairs the observed points with the integrated-intensity penalty $-\int p\,\lambda$.

II.8 Scale — coarsening, transport, and averaging (all phases)¶

This is where §I.2’s “point versus area depends on the instrument” becomes mathematics.

II.8.1 Spatial coarsening¶

Define the coarsening operator for pixel size $\Delta x$:

$C_\Delta$ maps a measure to a vector of pixel totals in $\mathrm{kg\,h^{-1}}$. Two facts:

1. Coarsening forgets the atomic/diffuse split. $\mu(P_j) = \mu_{\mathrm{pt}}(P_j) + \mu_{\mathrm{df}}(P_j)$ is one number; a pixel containing twenty $20\ \mathrm{kg\,h^{-1}}$ pads is indistinguishable from a pixel with $400\ \mathrm{kg\,h^{-1}}$ of diffuse flux.

2. A point source is an atom relative to $\Delta x$. Source $k$ is “point” for an instrument if its physical extent $\ll \Delta x$ and it is the dominant contributor to $\mu(P_j)$ for its pixel; otherwise it is part of an aggregate.

+---+---+---+---+
|   | o |   |   |   atoms resolve,
+---+---+---+---+   each in its
|   |   |   | o |   own pixel
+---+---+---+---+
→ point-process model

+---------------+
|  o    o       |   atoms merge
|     ::::::    |   with diffuse
|   :::::::     |   into one total
+---------------+
→ field model of C_Δ μ

II.8.2 Atmospheric transport — the field-side observation operator¶

The satellite does not retrieve $\mu$; it retrieves the raw column $y(x) = \mathrm{XCH_4}(x)$. The link is a transport operator

where $\mathcal{T}$ convolves emissions with a plume kernel driven by wind $U$, and $f_{\mathrm{bg}}$ is background. The enhancement product of §I.4 has the background already removed, $f = y - f_{\mathrm{bg}} = \mathcal{T}\mu + \varepsilon_r$. Model one or the other, never $f_{\mathrm{bg}}$ twice.

• For a single atom, $\mathcal{T}(s_k\delta_{x_k})$ is a compact plume of peak enhancement roughly $\propto s_k/U$.

• For a diffuse patch, $\mathcal{T}\mu_{\mathrm{df}}$ is a broad, low enhancement.

Per-plume rate estimation (IME and cross-sectional methods) is the local inversion of $\mathcal{T}$ for one atom; it returns a noisy $s_k\eta_k$ with $\eta_k$ log-normal (§II.9.3). In plumax the forward side of $\mathcal{T}$ is the Tier I–III dispersion models (Gaussian plume/puff, Eulerian FV).

II.8.3 Temporal scale — averaging and sampling¶

Two time scales matter: the instrument revisit $\Delta t$, and the source’s own on/off timescale.

Time-averaged rate. Over a window much longer than the on/off cycle,

This is the quantity an inventory wants; $s_k$ (the on-state rate) is what a single fine-imager scene measures.

II.9 Observing point sources — thinning and the full operator (Phases A, B)¶

II.9.1 Thinning¶

Definition. Given a point process $N$ with points $\{x_k\}$ and a retention function $p: S \to [0,1]$, keep each $x_k$ independently with probability $p(x_k)$. The kept points form $N_{\mathrm{obs}}$.

II.9.2 Rate-dependent thinning and the detection limit¶

Let the detection probability depend on the rate, writing $p(x,u)$ for a source at $x$ with rate $s = s_0 u$. A CRM with intensity $\lambda(\mathrm{d}x)\rho(\mathrm{d}u)$ is observed as

Take $p(u)$ to be the POD curve, rising around $s_{\min}/s_0$, and require

Under that condition:

1. $\rho_{\mathrm{obs}} = p\rho$ has finite mass: you observe finitely many sources. “Infinitely many” was about the truth, never the data.

2. The observed total is biased low by the factor

   $$\frac{\int u\,p(u)\,\rho(\mathrm{d}u)}{\int u\,\rho(\mathrm{d}u)}$$

   (24)

   This is the below-detection-limit gap, in one line.

II.9.3 The full observation operator for point sources¶

Each stage preserves Poisson-ness (thinning theorem; displacement theorem; superposition), so the whole operator is tractable.

Three differences from a data-assimilation $\mathcal{H}$:

1. Random, not deterministic.

2. Dimension-changing.

3. Not invertible. Only the product of the four thinning probabilities with $\lambda$ is identifiable. Separating “source off”, “not overhead”, “cloudy”, and “too small” requires each $p$ calibrated outside the data: controlled releases, orbit files, cloud masks.

II.10 Time (Phases B, C)¶

Nothing above required $X$ to be spatial. Time adds order.

II.10.1 Cumulative view — subordinators¶

Model discrete releases at $t_k$ with masses $m_k$ [kg]:

$M$ is always nondecreasing. It is a subordinator (independent, stationary increments, i.e. a CRM on the time axis) only if the pairs $(t_k, m_k)$ form a Poisson random measure with time-homogeneous intensity $\mathrm{d}t\,\rho(\mathrm{d}m)$, so that release times carry no memory and no seasonality. Under that assumption: Poisson gives a counting process; Gamma gives the Gamma subordinator; stable gives rare huge jumps.

Renewal, Hawkes, and time-varying intensities (§II.10.2) break the assumption, and then $M(t)$ is not a subordinator.

II.10.2 Conditional intensity — history dependence¶

Poisson: $\lambda^*$ ignores $\mathcal{H}_t$ (memoryless). Others:

One likelihood covers all:

II.10.3 GPs in time are SDEs — the Kalman link¶

Matérn GPs over $t$ with half-integer smoothness $\nu = p + \tfrac12$ are exactly linear SDEs with a $(p+1)$-dimensional state. General $\nu$ has no finite state-space form. The simplest case, $\nu = \tfrac12$, with $W$ standard Brownian motion:

(Ornstein–Uhlenbeck). For these kernels GP smoothing becomes a Kalman filter + smoother, $O(n)$. Squared-exponential kernels (and Matérn with non-half-integer $\nu$) get no exact such structure, only approximations.

II.10.4 Space × time¶

$X = S \times T$. Kernels are separable ($k_S \cdot k_T$) or non-separable; the advective kernel depends on the displacement $x - x' - \mathbf{v}(t - t')$ with velocity vector $\mathbf{v} = 86.4\,\mathbf{U}\ \mathrm{km\,d^{-1}}$, a plume carried by the wind in the wind’s direction. Per-source rate functions $s_k(t)$: the continuous-time version of §II.4.4 is a two-state Markov (or semi-Markov, for non-exponential durations) switching process. This is the right model for a leak that stays on until repair: the on-state duration is the repair time.

II.10.5 Clustering over time¶

Dependent DP: stick-breaking with time-varying $V_k(t)$, so attribution fractions drift smoothly instead of being re-estimated monthly.

II.10.6 The observation clock is a thinned point process¶

Overpass times form a point process on $T$ (deterministic per satellite, near-Poisson for a constellation), thinned by cloud. That is the “clock” and “clear sky” stages of §II.9.3, seen from the time axis.