A Non-Detection Is Not Zero

The uncomfortable gap between “not seen” and “not there”

Suppose an aircraft surveys a facility and reports no methane source. It is tempting to record a zero. The database needs a number; zero looks precise; and the flight produced no quantified plume.

But the instrument did not observe the emissions rate directly. It performed a detection experiment under particular conditions. Small sources are harder to see than large ones. Wind disperses a plume. Survey altitude changes the signal. A pass can also be unusable for reasons that have nothing to do with the facility.

So no report is an observation, not an emissions value.

The distinction matters whenever we use intermittent surveys to estimate emissions over a month, a year, or an entire portfolio. Treating silence as zero systematically removes plausible emissions paths before the analysis has even begun.

Detection is a probability, not a threshold

Let $Q$ denote an emissions rate, let $x$ collect the conditions of a survey, and let $s$ identify the sensor system and algorithm version. We will write that system’s detection probability as $p_s(Q,x)$:

For the interactive example below, the probability of detection is the published Bridger Photonics Gas Mapping LiDAR fit in Table 2 of Conrad, Tyner, and Johnson (2023). The authors derived continuous POD functions from controlled-release data. The fitted form is

where $h$ is aircraft altitude above ground level in metres and $u$ is wind speed at 3 metres above ground in metres per second.

This is not a universal law of methane detection, nor a generic curve for every aerial system. It is a technology-specific empirical fit: the same emissions rate can be easy to detect in one survey and easy to miss in another.

A reported rate has a likelihood too

Detection answers whether the system returns a report. Quantification answers what value it returns given that a detection occurred.

Let $Y$ be the reported emissions rate. A conditional quantification model has the form

For the GML teaching example, the published controlled-release results are represented with a multiplicative lognormal model:

The logarithm matters because quantification error is naturally multiplicative. A report might be half the true rate or twice the true rate; an additive “plus or minus a fixed number of kilograms per hour” does not describe that behavior well across small and large sources.

Detection and quantification form one observation model. If a rate $y$ is reported, the evidence for a proposed true rate is

The conditioning on $D=1$ is not decorative. Near a detection limit, the cases that become reports can be a selected subset of all measurement attempts. A quantification model calibrated only on detected cases must be paired with the detection process and reporting protocol under which those cases were selected.

Silence has a likelihood

If an evaluable survey produces no report, the likelihood of that outcome under a proposed emissions rate is

That quantity is rarely zero. Under representative conditions of 175 metres altitude and 3.5 metres per second wind, the example kernel gives very different interpretations to the same silent pass:

One non-detection barely challenges the 1 kg/h hypothesis. It weighs strongly against 5 kg/h. The observation is identical; its evidential force depends on the hypothesis being evaluated.

The third chart in the interactive simulator shows the conditional distribution of reported rates. Move the quantification log-error slider to widen or narrow that distribution. The reset value reproduces the paper-derived spread; other values are sensitivity experiments. Notice that a precise-looking reported number is still compatible with a range of true emissions hypotheses.

Bayes’ rule does the bookkeeping

Before the survey, we have some distribution over plausible emissions rates, $p(Q)$. After a non-detection, that distribution becomes

The update does not flip an emitting source into a non-emitting source. It reweights possibilities. Rates that were very likely to have produced a report lose probability mass. Rates that could plausibly have remained unseen lose much less.

This is the modest but powerful idea behind the second chart in the interactive model: the lime curve is not a new measurement. It is the old set of beliefs after asking which hypotheses are compatible with what the survey actually did.

Repeated silence can become strong evidence

If repeated passes are conditionally independent and share the same detection probability, then $k$ consecutive non-detections have likelihood

At the representative conditions above, three silent passes remain compatible with 1 kg/h about 47.9% of the time. For 2 kg/h, the probability falls below 1%. For 5 kg/h, it is effectively zero.

The independence assumption deserves scrutiny. Weather, flight design, instrument state, and persistent plume geometry can couple observations. In production inference, those conditions should be represented pass by pass rather than hidden behind a convenient exponent. The simplified expression is still useful because it reveals the mechanism cleanly.

“Not evaluable” is a different observation

An evaluable pass with no report says, “the instrument had a chance to see this source and did not.” A non-evaluable pass says, “this survey cannot tell us.” Those are not equivalent.

In a likelihood calculation, a non-evaluable pass contributes a factor of one:

It leaves every emissions hypothesis exactly where it was. Converting that record to a non-detection would manufacture evidence against larger sources. Converting it to zero would go further and manufacture an emissions measurement that never occurred.

From a pass to an annual estimate

An annual estimate must consider more than the rate at a survey instant. Sources begin and end between observations. Their rates can vary. Some events are detected; others are not. The estimator therefore has to integrate over many possible histories that could have produced the observed record.

Detection probabilities connect those hidden histories to the surveys we actually see. They allow a silent observation to constrain a path without erasing it. Arachne is being developed around that broader inference problem: combining intermittent observations, externally calibrated measurement models, and explicit uncertainty to estimate emissions through time.

The important point is not that uncertainty disappears. It is that uncertainty becomes traceable.

The practical rule

When a survey produces no report, do not ask, “What zero should I enter?” Ask:

1. Was the survey evaluable?
2. What was the probability of detection under the conditions of that pass?
3. Which emissions hypotheses remain compatible with the silence?
4. How should this evidence combine with the rest of the time series?

A non-detection is not nothing. Used carefully, it is evidence—with a strength we can calculate.

Reference

Bradley M. Conrad, David R. Tyner, and Matthew R. Johnson, “Robust probabilities of detection and quantification uncertainty for aerial methane detection: Examples for three airborne technologies,” Remote Sensing of Environment 288 (2023): 113499.