Tier V.C — Persistency - Research Notebook

Question: Given the inverted intensity $\lambda(t)$ from Tier V.B, when will the next emission event happen, and what’s the probability of an event during a specified window?

This is the operational layer — what an LDAR (Leak Detection and Repair) crew or a satellite-tasking dispatcher actually consumes. The full derivations of each metric live in methane_pod/notebooks/08_persistency; this page summarises the metrics and how they slot into the plumax API.

The four operational metrics¶

1. Expected wait time $\mathbb{E}[\Delta t \mid t_0]$ ¶

How long after time $t_0$ until the next event?

\mathbb{E}[\Delta t] \;=\; \frac{1}{\lambda_0} \qquad \text{(homogeneous Poisson; memoryless)}

((1))

\mathbb{E}[\Delta t \mid t_0] \;=\; \int_{t_0}^{\infty} \exp\!\left(-\int_{t_0}^{t} \lambda(u)\, \mathrm{d}u\right) \mathrm{d}t \qquad \text{(inhomogeneous; depends on starting clock)}

((2))

For a diurnal source, vastly different at noon vs. midnight.

Operational use. Dispatch decisions: arrive during a high-λ window and the next event is imminent (worth waiting); arrive during a low-λ window and you’d waste hours. Drives MARS-style dispatch suppression during dormant cycles.

2. Probability of occurrence $\mathbb{P}\!\bigl(N(t_1, t_2) \geq 1\bigr)$ ¶

What’s the chance of at least one event in $[t_1, t_2]$ ?

\mathbb{P}\!\bigl(N(t_1, t_2) \geq 1\bigr) \;=\; 1 - \exp\!\bigl(-\lambda_0\, (t_2 - t_1)\bigr) \qquad \text{(homogeneous)}

((3))

\mathbb{P}\!\bigl(N(t_1, t_2) \geq 1\bigr) \;=\; 1 - \exp\!\left(-\int_{t_1}^{t_2} \lambda(t)\, \mathrm{d}t\right) \qquad \text{(inhomogeneous)}

((4))

Operational use. “Wrench-turning” probability. If a maintenance window is 4 hours, what’s the chance the leak shows itself during that window? Drives whether to schedule the visit.

3. Conditional intensity given prior detection $\lambda(t \mid t_\text{prev})$ ¶

For a source with a known recent detection at $t_\text{prev}$ , what’s the posterior intensity going forward?

For Poisson processes (no memory): unchanged. For Hawkes / self-exciting processes: bumped —

\lambda(t \mid t_\text{prev}) \;=\; \mu + \alpha\, \exp\!\bigl(-\beta(t - t_\text{prev})\bigr)

((5))

— captures the empirical observation that super-emitters “cluster”.

Operational use. Prioritisation: a source with a recent detection is more likely to repeat-emit in the next 24 h. Re-task a high-resolution satellite (GHGSat GHGSat Inc., 2016, Carbon Mapper Carbon Mapper, 2024) on top of a TROPOMI alert Veefkind et al., 2012.

4. Cumulative event count $\mathbb{E}[N(0, T)]$ and credible bounds¶

Expected number of events in $[0, T]$ , with credible interval from the posterior on λ.

\mathbb{E}[N(0, T)] \;=\; \Lambda(T) \;=\; \int_{0}^{T} \lambda(t)\, \mathrm{d}t

((6))

(Homogeneous: $\lambda_0 \cdot T$ .)

Operational use. Annual reporting, regulatory compliance. “How many emission events should we expect this year at this facility class, with 95% credible interval?”

API shape¶

A thin wrapper around methane_pod.intensity:

from plume_simulation.population.persistency import (
    expected_wait_time,
    occurrence_probability,
    cumulative_count,
    next_event_quantile,
)

# Inputs: posterior samples of intensity parameters (from Tier V.B fit)
# Outputs: posterior samples of the operational metric

E_wait = expected_wait_time(intensity, t0=18.0, posterior_samples=mcmc.get_samples())
# → array of shape (n_samples,) in [hours]

P_occur = occurrence_probability(intensity, t1=8.0, t2=12.0,
                                 posterior_samples=mcmc.get_samples())
# → array of shape (n_samples,) in [0, 1]

The metric functions take an intensity callable (any of the 13 equinox modules from methane_pod.intensity), a query window, and a posterior sample of the intensity’s parameters. They return posterior samples of the metric — full UQ propagation, no point estimates.

Module layout¶

Table (1):Tier V.C module layout — concern, target module, status.

Concern	Module	Status
Intensity functions	`methane_pod.intensity`	✓
Wait-time / occurrence / cumulative metrics	`plume_simulation.population.persistency`	☐
Posterior-aware metric wrappers	same module	☐
Operational dashboard / report templates	out of scope for `plumax`; lives in `plumax-deploy` (future)	—

The integral over $\lambda(t)$ in the wait-time formula is closed-form for a few intensity choices (constant, exponential decay) and otherwise needs jax.scipy.integrate or a fixed quadrature. Worth wrapping once and reusing across metrics.

Validation strategy¶

Homogeneous limit. For constant λ, all four metrics have closed-form formulas; the implementation should match to machine precision.
MC self-consistency. Sample $n$ event times from a known $\lambda(t)$ via thinning, compute the empirical wait time / occurrence frequency, compare to the closed-form metric. Tests both the metric implementation and the simulator.
Posterior coverage. For a synthetic source with known $\lambda_\text{true}(t)$ , the 95% credible interval on $\mathbb{E}[\Delta t]$ should contain the truth ~95% of the time across replicates.
Diurnal sanity. A solar-heated tank with peak λ at 14:00 should have $\mathbb{E}[\Delta t \mid 14{:}00] \ll \mathbb{E}[\Delta t \mid 02{:}00]$ . Numerical sanity check, not a formal test, but catches sign errors.

Open questions¶

References¶

GHGSat Inc. (2016). GHGSat WAF-P imaging spectrometer constellation. https://www.ghgsat.com/
Carbon Mapper. (2024). Carbon Mapper: airborne and satellite imaging spectroscopy for greenhouse gas monitoring. https://carbonmapper.org/
Veefkind, J. P., Aben, I., McMullan, K., Förster, H., de Vries, J., Otter, G., Claas, J., Eskes, H. J., de Haan, J. F., Kleipool, Q., & others. (2012). TROPOMI on the ESA Sentinel-5 Precursor: a GMES mission for global observations of the atmospheric composition for climate, air quality and ozone layer applications. Remote Sensing of Environment, 120, 70–83.

Tier V.C — Persistency

The four operational metrics¶

1. Expected wait time E[Δt∣t0]\mathbb{E}[\Delta t \mid t_0]E[Δt∣t0​]¶

2. Probability of occurrence P ⁣(N(t1,t2)≥1)\mathbb{P}\!\bigl(N(t_1, t_2) \geq 1\bigr)P(N(t1​,t2​)≥1)¶

3. Conditional intensity given prior detection λ(t∣tprev)\lambda(t \mid t_\text{prev})λ(t∣tprev​)¶

4. Cumulative event count E[N(0,T)]\mathbb{E}[N(0, T)]E[N(0,T)] and credible bounds¶