Journal of Optics
J. Opt. 22 (2020) 103001 (18pp)

https://doi.org/10.1088/2040-8986/abb1cf

Survey of autonomous gas leakdetection and quantification withsnapshot infrared spectral imaging
Nathan Hagen
Utsunomiya University, Department of Optical Engineering, Center for Optical Research and Engineering(CORE), Utsunomiya, Japan
E-mail: nh@hagenlab.org
Received 19 April 2020, revised 5 August 2020
Accepted for publication 24 August 2020
Published 22 September 2020
Abstract

We survey the current state of the art in industrial gas sensing, explaining how existing sensing
techniques estimate gas concentration, column density, total volume, and leak rate. We
especially focus on the application of snapshot infrared spectral imaging to gas detection andquantification, as this is the newest of the available techniques. After discussing the strengths
and weaknesses of each measurement technique for deployment in autonomous sensing, we
explain how autonomous detection systems are important to improving sensing capabilities inindustrial facilities, and advancing the state of the art.
Keywords: gas detection, infrared imaging, leak rate, emissions quantification, autonomous
sensing(Some figures may appear in colour only in the online journal)

1. Introduction

long-term monitoring of equipment and environmental conditions, detecting trespassers, and more.
For an autonomous system to be useful, it must have a low
false detection rate, be reliable at alerting when a dangerous
situation occurs, and should be effective at quantifying gas
plume sizes and leak rates—all without human involvement.
In the discussion below, we explain how snapshot infrared
spectral imaging is the key to making this approach workable. In order to show how autonomous instruments can beused to overcome existing labour-intensive methods, we survey the capabilities and limitations of this rapidly developing
approach and compare with alternative instrumentation. After
reviewing the theory for detection and quantification of gas
clouds using passive spectral imaging, we show measurementsfrom early field deployments.

Two important challenges face the oil and gas industry: an
increasing number of explosions at industrial facilities caused
by gas leaks [1], and tightening controls on gas emissions.
These challenges point to a need for improved equipment
and methods for gas leak detection and quantification [2],not only for locating fugitive emissions (caused by malfunctioning parts) but also for quantifying emissions by equipment that release gas as part of their design, such as flaresand pressure relief valves [3]. Current leak detection andrepair (LDAR) protocols as well as emission quantification
efforts are implemented by human engineers, so that evaluation is time-consuming and expensive. This high cost of current methods presents a barrier to improvement.
Autonomous gas sensing methods can advance the current
state of affairs by removing humans from the most labourintensive detection and monitoring tasks. Autonomous systems can operate continuously night and day, can log andtransmit measurements in real time, and have been demonstrated to not just detect leaks but quantify them at the sametime. Moreover, they can provide ancillary information such as
2040-8986/20/103001+18$33.00

2. The gas detection status quo
At chemical processing facilities, the standard protocol tomonitor for gas leaks in the United States is EPA Method 21[4], which requires that designated components in the facility(valves, flanges, pumps, etc) be leak-tested at regular intervals.
1© 2020 IOP Publishing Ltd

Printed in the UK

J. Opt. 22 (2020) 103001

N HagenAn LDAR engineer implementing the monitoring procedure
carries a portable gas leak detection instrument (a ‘gas sniffer’)and slowly passes the instrument’s probe head around the component. The instrument draws ambient air up through the probehead and into a measurement chamber that quantifies the gas
concentration in the air. For gas leaks whose measured concentration exceeds a given threshold for the component under
test, the protocol demands that the component be scheduled
for repair.
While the Method 21 procedure is straightforward, we canalso see that the subjective behaviour of the engineer plays asubstantial role in the outcome. For instruments with a slow
response time, the engineer must be careful to hold the probehead at the source of the leak for a time period exceeding
the instrument’s 90% response time. Meanwhile, wind conditions and component accessibility affect the extent to which
the engineer can locate and accurately quantify the leak.
Method 21 also mentions an alternative procedure in which
a soap solution is sprayed onto the components under test, a
procedure sometimes referred to as ‘soaping’. The user thenlooks for the appearance of bubbles. If none appear, then we
presume that there is no significant leak; if bubbles appear thenthe protocol calls for returning to the location of the leak witha gas sniffer.
For quantification of the volumetric flow rate of a leak,
Method 21 suggests two approaches [5]. The more accurate
approach is to seal the leak source with a plastic bag and allow
the trapped gas to pass through a flow meter and a gas concentration detector, together which can be used to give theleak rate in units of L/min or g/s. While this ‘bagging’ procedure provides an accurate measure of the size of the leak, it
is extremely labour intensive, so that an engineer can quantify
only about 10 leaks per day this way. Method 21 gives a second
leak rate quantification method, which is simply to multiply
the measured gas concentration near the leak, using established scaling factors that vary according to the item under
test. The result, however, can vary over three to four ordersof magnitude [6, 7], so that the quantification uncertainty inthis latter approach remains frustratingly large.
Recognising the advantages of using infrared cameras forgas detection, facilities have increasingly turned to infrared
imaging for LDAR monitoring. This process involves briefly
staring at the components under test with an infrared camera
and looking for any signs of gas leaks. Since a user can examine many components within a single image, can image components that are difficult to access with a sniffer, and can evaluate a component for leaks more quickly than with a sniffer, it isgenerally acknowledged that optical gas imaging (OGI) leakdetection is about 10 times more efficient than sniffer-based
detection [8].
In 2008, the EPA approved an alternative work practice,effectively an amendment to Method 21, allowing the use of
OGI in place of Method 21’s use of gas sniffers [9, 10]. In
2016, the EPA took this even further and issued new rules[11] requiring the use of OGI as part of a facility’s LDAR
protocol. However, while infrared cameras have proven useful in detecting leaks, their use in quantifying leaks has only
recently been analysed, and is the subject of ongoing research.

Figure 1. An example detection using an infrared camera viewing a

propylene gas release, with the gas column density overlaid onto avisible camera image.

What we show in sections 6 and 7 below is that quantification with infrared imaging is now viable, but that users—
and algorithms—need to be better trained to understand how
to achieve the conditions necessary for good results. Finally,
while the algorithms and methodology are still too new tomake their way into the EPA’s rules, researchers are busy
developing methods to quantify gas leak rates using infrared
camera video data.
While the discussion above has centred on regulation developments within the United States, European regulatory agencies are following a similar trend, with several countries stillrequiring sniffing, and other countries allowing either OGI or
sniffing to satisfy monitoring requirements [12].

2.1. Emissions monitoring

In contrast to the detection-centric goal for safety LDAR,
emissions monitoring focuses on quantifying gas emissions.
These two goals are of course closely connected, since one
cannot quantify what one cannot detect, but they are also lessclosely related than one might at first expect. For autonomous detection, an essential part of the task is to keep false
detections to a minimum, and for this it is necessary to use
tight constraints on the detection algorithm. For a system aiming for accurate quantification, however, false detections are alesser concern if they do not have a strong impact on the total
amount of gas detected. A system can often improve emissions measurement accuracy by using looser constraints on thedetection algorithm. A second difference in the two modes ofoperation is that emissions monitoring takes place over a much
longer timeframe than will an algorithm focused on safety andlocating leaks. Thus, monitoring timescales are often days ormonths as opposed to the second- or minute-response times ofsafety monitoring.
When quantifying gas emissions, users are often attempting to characterise the efficiency and environmental impact ofoperations. For safety and LDAR, intentional emissions are2

J. Opt. 22 (2020) 103001

N Hagentreated very differently than fugitive emissions, but for emissions monitoring both categories are treated on an equal footing. The difference is significant because the two categories
have been shown to be similar in magnitude for industrial
chemical processing plants. [13]Emissions monitoring suffers from an even larger technology gap than gas leak detection does. Because existing tools
for estimating the quantity of gas emissions are largely humandriven, it is labour-intensive and costly to monitor variable
and intermittent emissions sources. The recent literature on
emissions from natural gas production, for example, contains a
lot of hotly contested numbers, stating that conventional EPA
estimates [14] are excessive [15], about right [16], or underestimate total emissions [17, 18], depending on the source [19].
While a significant portion of the controversy arises from differences between ‘bottom-up’ and ‘top-down’ measurement
methods, in which top-down methods typically estimate muchlarger emissions, it is clear that better measurement methods
are needed for resolving these estimates. While infrared imaging cameras have had a significant impact on reducing fugitive emissions, their impact on emissions quantification so far
has been limited.

that are robust enough for users to depend on them to makeimportant decisions.
There are four types of sensors currently used for gas detection and quantification: point sensors, line sensors, infrared
imagers, and snapshot infrared spectral imagers. While eachtechnology has its own strengths and weaknesses, such that an
autonomous detection and monitoring framework that combines multiple sensor types seems likely to develop, snapshot
infrared spectral imaging is an essential tool to make autonomous approaches practical and effective.
3.1. Point sensor networks

Point sensors are gas detectors that analyse gases passing directly through them, with the most common sensor types being
chemi-resistive and infrared [20]. Chemi-resistive sensors use
a film whose electrical resistance changes in response to chemical environment changes, while infrared optical sensors measure the absorption signature of gas passing through a smallchamber. Point sensors themselves and their associated electronics are typically quite small, but satisfying the strict safety
specifications required by many industrial facilities means thatthey are packaged in durable explosion-proof casings thatmakes them larger and heavier than one might expect.
Point sensors report the detected gas in concentration units(parts per million, or ppm), have typical response times on theorder of seconds to tens of seconds, and can often detect concentrations of < 1 ppm, depending on the technology used andthe type of gas detected [21, 22]. They typically cannot discriminate among different gas types, and generally have different response amplitudes for different gases, so that the gas
type must be known a priori for accurate concentration measurement. Point sensors used in fixed emplacements detect thegas passing through their immediate vicinity, while sensors
attached to workers’ uniforms detect gas present in the area
through which a worker walks. They have also been attached
to robots or moving platforms [23].
Because point sensors detect the gas concentration at one
location, by themselves they cannot distinguish between a lowlevel leak nearby, a large leak far away, or even the directionof the leak. Industrial chemical processing facilities have point
sensors distributed at key locations, such as in figure 2, so it
is natural to try networking the sensors together to localiseleaks. With multiple sensors operating as a network, localising
the leak becomes more effective as the density of the network
nodes increases. The complex 3D behaviour of gas plume andnatural wind motion, however, make this a challenging inverse
problem. Even at short distances from a leak, point sensor
detections can be erratic. Averaging the detected concentration over longer measurement periods effectively smooths out
the most complex details of the motion, improving the problem’s invertibility [24–26], but any sensor layouts near leaksources still require fluid dynamical models to determine therelationship between air flow patterns and detected concentrations. Chraim et al, for example, demonstrated experimentally how a sensor grid with nodes spaced 4 m apart allowed
for localising leaks to within 5 m of the source, for measurements averaged over a 100 s period [27]. If we consider an

2.2. Leak rate estimation

The third mode of operation in gas sensing is estimating thevolumetric flow rate of a leak, in which an engineer wants tocharacterise the size of an already detected leak in flux units(volume per unit time) such as L/min or g/s. While leak rateestimation is closely connected to emissions quantification, it
differs from the other modes in that current technology fordoing this is already quite accurate. The problem to solve,
therefore, is not to improve the accuracy of existing methods
but to make them more practical and less labour intensive.
The established method for measuring leak rate is to wrap
the area of the leak in an air-tight enclosure (‘bagging a leak’)and connect a flow rate meter and gas concentration sensor to
the enclosure outlet. The problem with this standard method,
however, is that it is labour intensive, and it also directly
exposes the engineer to the leaking gas, which is a safety problem. An automated solution that can avoid the safety hazard isa goal that the industry has been pursuing.
3. Autonomous methods for gas sensing andquantification
Although infrared imaging has been shown to be a more efficient method for LDAR than sniffing, it remains dependent on
the person viewing the camera display to evaluate whether or
not a gas leak is present. Thus, these methods are still problematic for tasks such as continuous monitoring or detectionof intermittent leaks. Autonomous methods do not rely directly on human labour, and thus allow continuous monitoring capabilities that would be prohibitive with existing protocols. This can involve using existing sensors in new ways, suchas installing sensors on mobile robot platforms, or developing
entirely new sensor types, and require measurement methods3

J. Opt. 22 (2020) 103001

N HagenFigure 2. Conceptual image of a point sensor network at anindustrial site, where each red circle indicates the location of asensor.

Figure 3. Gaussian plume model.

example facility that is 1 km2 in area, however, this spacing
requires a network of 62,500 sensors, and that is only for thetwo-dimensional area of the facility and so ignores the need forsome vertical stratification of nodes as well, since most gases
are significantly lighter or heavier than air.
The most common way of getting out of these measurement
difficulties is to change the scale of the model in both space
and time. By averaging over long periods of time, and by analysing the concentration far downwind of the leak—far awayfrom the disturbing effects of nearby equipment on airflow—
the time-averaged behaviour of the gas mixing with the ambient air comes to approximate a kind of diffusion transport
under certain atmospheric conditions. Diffusing fluids havewell-behaved Gaussian distributions of concentration that areeasily inverted by estimating a small set of model parameters.
This is the Gaussian plume model of gas emission [28].
In a Gaussian plume model, the 3D gas concentration distribution ρ(x, y, z) downwind of a constant leak source can beconstructed for either smooth wind flow or turbulent flow, byincorporating different diffusion constants [29, 30]. In smoothwind flow, ρ represents an instantaneous snapshot of the gas
concentration, while in turbulent flow ρ represents a time average of the gas concentration, with the decorrelation time of theplume flow giving the minimum length of time to average over.
Appropriate time scales range from 10 min to 10 hr dependingon the extent that turbulence causes changes in wind direction [25, 31]. Figure 3 shows a plume model overlaid on an
example gas release.
A widely-used version of the Gaussian plume model gives
the downwind gas concentration ρ (g m−3 ) by [31]

ρ(x, y, z) =

[
]2
(
z−h )
ϕ
y2
exp − 2 −2πuσy σz2σy2σz2model of (1) does not account for gas buoyancy, it can easily
be adapted to include it [32].
With the Gaussian plume model and knowledge of theinstantaneous (for smooth flow) or time-average (for turbulent flow) wind speed u and direction together with the general atmospheric conditions, we can take any three or moregas concentration measurements from point sensors within
the plume to determine the (x, y, z) location and intensity ϕof a leak from the Gaussian parameters h, σ y , and σ z . While
this means that only three sensors, or a single sensor transported to three locations, are in principle needed to localise
and quantify a leak, the difficulty of accurately satisfying themodel assumptions and the extended time required to averageout turbulence effects can be problematic. Attempts to verify
measurement accuracy have run into difficulties with model
specification (e.g. defining the dependence of σ y , and σ z on
distance x) and the need to accurately measure low concentration values so far from the leak source [33, 34].
As implied by figure 3, the Gaussian plume model is only
appropriate when the air flow is unobstructed, so that themodel is not useful for locating leaks within the perimeter ofcomplex industrial facilities but only far from leak sources.
Large-scale plume vortices may even dominate dispersion ofplumes 5 to 10 km downstream of the source [35]. Closer
to the leak source, more accurate fluid dynamical models of
the plume are needed, constructed from knowledge of the 3D
shape of the structural environment [22, 36].

3.2. Line sensors

Line sensors are the second type of gas detection technology.
A typical line sensor system uses a laser light source and a photodiode detector, with the laser tuned to a wavelength matching a strong absorption peak of the target gas. The detector
commonly uses a narrowband spectral filter so that it is insensitive to any light other than the illumination laser. For gas
detection, the laser passes along a line through the region to
be monitored, at the end of which is either the detector or aretroreflector that returns the laser back towards the detector(i.e. bistatic or monostatic configurations) [37].

(1)

where x is the distance downwind from the leak source, z is thevertical displacement from the ground, ϕ is the gas leak rate(g/s), and u is the x-direction wind speed (m s−1 ). The plumeconcentration profile width parameters are given by σ y (x) andσ z (x) in the horizontal and vertical directions, while h is thevertical position of the middle of the plume. While this simple
4

J. Opt. 22 (2020) 103001

N Hagen

3.3. Gas imaging techniques—active sensing methods

The third gas sensing technology is gas imaging, implementation methods for which can be separated into two broad categories of active and passive techniques: either requiring a specialised light source or using the natural environment for measurement [39]. Like line sensors, active imagers generally use
a laser for illumination, but scan the illumination beam acrossthe field of view in order to estimate the spatial distribution ofgas. Thus, many are scanning-based extensions of line sensing
methods.
Differential absorption LIDAR (DIAL) uses a pulsed tunable laser operating at two discrete wavelengths, one at astrong absorption peak and the other at an absorption minimum. Using the pulses for time-gating, the DIAL system
looks for the backscattered signal and uses the return time todetermine the distance to the gas. Thus, unique among the gas
imaging techniques discussed here, DIAL’s unit of measureis´
the gas concentration ρ rather than its integral ζ = ρ(z) dz.
An example DIAL instrument [40] was used to measure thegas concentration in a 50 m× 500 m vertical plane over a
period of 5–7 min.
Backscatter absorption gas imaging (BAGI) illuminates ascene with laser light at a wavelength tuned to an absorption peak of the target gas, analysing changes in backscattered
light from the scene for presence of gas [41, 42]. When thelaser wavelength is strongly absorbed by the target gas, the gas
plume thus appears as a black cloud against the more brightly
illuminated equipment background. This makes for a compact
imaging instrument, but suffers from reduced light intensity,and therefore sensitivity, in comparison to the line sensing
methods.
The scanning laser illumination methods of active gas
imagers means that they are generally too slow to provide
video-rate measurements, even when using high-power laser
sources. As a result, they have difficulty measuring the wisps
and swirls of turbulent gas that occur at small scales, so they
they are best suited to large-scale leaks or smooth flow conditions.

Figure 4. Three types of gas sensing instruments: point sensor, line

sensor, and imager.

A modified implementation of the line sensor involves
using two lasers, or a single laser emitting two wavelengths,
where one wavelength is tuned to a gas absorption peak while
the other is tuned to an absorption minimum. This helps to distinguish between absorption by the target gas and absorption
by an interference agent such as steam or dust. Tunable diodelaser absorption spectroscopy (TDLAS) uses a similar technique of tuning the laser wavelength across the gas’ absorption
spectrum profile [38].
Line sensors have a significant advantage over point sensors
in that a single instrument can monitor for gas along all points
intersecting the laser, rather than at just one location. The
use of specific gas absorption wavelengths makes the line
sensors robust to false detections, while also maintaining ahigh sensitivity. A major difference between these instruments
and point sensors, however, is that they use a less familiar
unit of measure. Rather than gas concentration ρ (units ofm−3 ), the unit of measure is the gas column density ζ (units ofm−2 ) — the gas concentration integrated along the laser path
length,3.4. Gas imaging techniques—passive sensing methods

Passive imaging instruments use only the natural signals fromthe environment to detect gas plumes. The basic principle thatthey all employ is that when gas passes between the camera and the background scene, it reduces (or enhances) theradiance of the background if the gas is cooler (or warmer)
than the background [43–45]. This difference between gas andbackground radiance is generally referred to as the ‘thermal
contrast’ or ‘radiance contrast’. While this measurement principle has the advantage of not needing an artificial lightsource, it has the disadvantage that it must rely on sophisticated algorithms to separate natural changes in the scene
radiance from any changes due to gas. Sections 4–7 below
discuss the theory in detail and show example experimental
data.
Gas correlation imaging is a method that has been aroundfor over two decades now, and works by viewing a sceneˆζ=ρ(z) dz ,where z is the propagation direction of the laser. While thecolumn density units can take more effort to get used to thanconcentration does, ζ is actually more useful than ρ for leakrate quantification, as shown in later in section 6. (The symbolζ is appropriate for the column density in that its shape somewhat resembles both a ‘c’ and a ‘d’ for ‘column density’.)5J. Opt. 22 (2020) 103001

N Hagensimultaneously through one spectral filter and one transparent window, either using a split pupil or two separate cameras [46–48]. The spectral filter is constructed as a cylinder
filled with the target gas—an effective way to optimise measurement sensitivity, since the filter transmission function isexactly matched to the target gas. The camera behind the filter
experiences a larger relative change in signal when the target
gas appears than will the camera with the broadband filter, andthis difference can be used to calculate the column density ofgas present. Sandsten et al showed example 136 × 136 imagesof methane, ethylene, and ammonia captured at 15 Hz frame
rates [47].
Filtered-camera gas imaging has the simple hardware setup
of inserting a narrowband spectral filter in front of a cryocooled thermal camera [13]. While the narrowband filter is not
fully optimised to the target gas, as it is in correlation imaging,
this method requires only one camera/pupil and works for awide range of gases. A viewer watching the scene through oneof these cameras looks for changes in the scene radiance toindicate the presence of gas moving across a scene. The simplicity and effectiveness of this approach has made it the mostpopular choice for LDAR use, to the point that many now refer
to it as ‘smart LDAR’.
A disadvantage of the filtered-camera approach is that it is
not gas-specific, and thus can only be used for quantification if
the gas species is known a priori. An alternative is to allow thecamera to use one of a number of different filter bands [49].
Since gas spectra are more distinct in the longwave spectral
band (LWIR, 8–12 µm) than the midwave range (MWIR, 3–5µm) usually used by filtered cameras, this approach operates
in the former range, and employs a filterwheel to select among
a set of different spectral filters for a specific scene.
While these passive methods have a large speed advantage over active imaging techniques, they generally have difficulty in accurately estimating absolute column densities rather
than relative column densities. The measurements are typically obtained as the difference in radiance between the current scene and a reference scene. In order to estimate absolute column densities, these systems have to make assumptions about the gas present in the measurement and reference
scenes—assumptions which can make the measurements susceptible to error.
Imaging Fourier transform spectrometry (IFTS) uses
an imaging Fourier-transform spectrometer to measure theinfrared radiance spectrum at each pixel of the scene [50]. The
instrument spectral resolution is typically chosen as a tradeoffof spectral information detail for measurement speed, so thata user can reduce the speed of measurement in order to gethigh spectral resolution, and therefore higher sensitivity togas. A strength of this approach is that the detailed spectrum
allows an algorithm to estimate the quantities of multiple species of gases simultaneously. This makes measurements robust
against interference gases such as steam, and allows the quantification of gas mixtures, even when the gas mixture proportions vary from pixel to pixel. Sabbah et al demonstrated theuse of an imaging FTIR set to 4 cm−1 resolution over the1050–1150 cm−1 spectral range (i.e. 8.3 nm average resolution
over 8.70–9.52 µm) imaging methane emitted from industrial

stacks and also a large plume of SF6 gas viewed from a distance of 800 m [51]. At this resolution, however, the imaging
rate achieved was only 1 frame per 15 seconds (0.067 Hz).
3.5. Gas imaging techniques—snapshot spectral imaging

Snapshot spectral imaging is the newest among gas sensing
techniques. It consists of a method that augments the passive
infrared imaging techniques with recently-developed instrumentation to measure the infrared spectrum at each pixel ofthe scene in real time. [52] This maintains the ability to image
scene dynamics at video rate, while also keeping the robustness advantages of capturing the spectrum. Since smaller gas
plumes are inherently dynamic, with rapidly moving swirls
and eddies, this video measurement capability is essential
for better sensitivity and locating smaller leaks. Moreover,since many industrial facilities are permeated with steam,and with pipe structures placed in dust-prone environments,using the radiance spectrum to remove false detections due tothese interference agents provides a more robust platform forautonomous detection. Finally, as with IFTS, having the spectrum at each pixel in the scene also allows the user to discriminate between gas types and to measure gas mixtures.
The recent development of snapshot spectral imagers takes
advantage of the confluence of four modern technology
advances: fast computing, large-pixel-count detector arrays,
high-speed data transmission, and precision manufacturing.
Each of these advances has been essential to making the technology work, so that it is only in the past decade that snapshot
spectral imaging methods have become practical for industrial sensing. Although these systems can be difficult to build,they basically operate like N different gas imaging cameras,
all imaging the same scene at the same time, using N different
spectral transmission bands. Thus, unlike with scanning systems such as IFTS, the snapshot approach does not have tosacrifice light efficiency in order to collect spectral data, andcollects the data at a much faster rate, though at a lower spectral resolution than IFTS [53]. Additional advantages from analgorithmic point of view are that because the data is collected
in snapshot form, there are no motion artefacts due to scanning over a moving scene, and algorithms can make use of the
a complete 4D (x, y, λ, t) dataset for measuring spatio-spectrotemporal correlations to detect gas. For example, a gas plumeemanating from a leak will produce a distribution of gas near
the leak that will, in the next frame of the video, expand out
into the neighbouring pixels. Therefore the spatial elements
in one frame are closely correlated to nearby spatial elementsin the subsequent frame—a priori information that algorithms
can put to good use to improve detection sensitivity.
Currently the only snapshot spectral imaging systems actively deployed in industrial gas sensing are based on uncooled
longwave infrared detectors, so that they have different measurement characteristics than the midwave (3–5 µm) filteredcameras used in ‘smart LDAR’. Midwave cameras view the‘functional group region’ of gas absorption spectra, where
many hydrocarbon gases have similar spectral shapes. Cameras that work in the longwave (8–12 µm) spectral region,
however, operate in the ‘fingerprint region’ for these gases.
6

J. Opt. 22 (2020) 103001

N HagenIn this region, the spectral shapes of the various hydrocarbon
gases are more distinct from one another, so that speciation ismore robust.
Two driving factors determine that there will be large differences between practical snapshot spectral imager detection
and quantification algorithms and those of the existing literature: the need for autonomous measurement, and the need tooperate on a live video stream. The former precludes methods
that require user input and supervision, while the latter precludes any algorithm that cannot operate live on a data streamof > 5 megavoxels per second. In addition, the need to collect video-rate data in real time precludes the use of highresolution spectroscopy, as this both increases the data rateeven further and reduces the signal-to-noise ratio of the data.

Figure 5. The measurement geometry for a single pixel in a gas

cloud imager: the camera line of sight views the backgroundinfrared radiation (1) through a gas cloud layer (2) and a foregroundatmospheric layer (3). Each layer has a spectral radiance L,
transmission τ , and temperature T.

3.6. What each technology means for autonomy

passing between the camera and the background [60]. We canmodel the effect of a gas cloud passing across a scene with athree-layer radiative transfer system (figure 5), in which

Autonomous sensing for gas LDAR and emission monitoring
has long been a goal for instrumentation developers, but hasremained out of reach. For point and line sensors, the sensing regions are often too sparsely sampled to give sufficient
information for autonomous action. For infrared imagers, theprimary difficulty has either been the vulnerability to interferents such as steam, or failure to discriminate among gas
species. For scanning-based spectral imagers such as IFTS,
it has been difficult to overcome the low data rate and poor
SNR. Snapshot spectral imagers have have a much higher
light throughput [54] than traditional scanning methods canachieve, and allow for video-rate imaging [55]. Since gas
clouds are dynamic phenomena and require video analytics toproperly detect them, these are important advances for improving sensitivity, reducing false detections, and accurately quantifying the gas.
The discussion so far has focused on fixed emplacements ofpoint, line, and imaging sensors, but another option is to attachsensors to mobile platforms—vehicles, mobile robots, or facility workers themselves—so that the gas leak detection can be
pursued actively [56]. Mobile sensor platforms can use chemotaxis, the principle by which animals locate the sources ofscents for food etc to locate gas sources by moving in the direction of highest concentration gradient [27, 57]. Future facilities
may see a network of mobile detection robots deployed, eachmonitoring for leaks and, upon detection, collectively searching for leak sources.
Mobile options for LDAR have not yet seen much development, possibly due to the safety difficulties of operating
moving equipment in strict safety conditions and potentiallyflammable environments. However, a number of researchers have developed mobile platforms for emissions monitoring and quantification, such as by attaching a series of point
sensors to a truck that passes through a thin-concentration(non-inflammable) gas cloud [58, 59].

(a) Spectral radiance Lb (λ) is generated within a source region
which can be either an opaque object such as the ground,or the atmosphere itself, such as when viewing a cloudlesssky, or a combination of the two.
(b) The source spectral radiance traverses the gas cloud layer,
and is attenuated/increased by absorption/emission ofgases located there.
(c) The radiation passes through an atmospheric layer to reachthe camera.
This three-layer model does not require any assumptions on
the background spectrum Lb (λ) behind the gas cloud [61].
In the discussion below, we use the following variable
definitions:
Lf , Lg , Lb

spectral radiances originating from foreground, gas, and background layers
radiant flux at the camera pupiltransmission of foreground and gas layers
emissivity of foreground and gas layers
temperatures of foreground and gas layersMτ f, τ gϵf , ϵ g
Tf , Tg

The absorption spectrum of a gas is defined by[

αg (λ) = 1 − exp −≈ σ(λ) ρℓ ,

(ˆ ℓ

)]
ρ(z, λ) dz

0(2)

where ℓ is the path length through the gas cloud. The approximation used here assumes that the gas cloud is homogeneous(so that the integral becomes a simple multiplication) and thatthe absorption is small (α ≪ 1) so that the exponential can beapproximated as e−x ≈ 1 − x so that α ≈ x. This is the ‘thingas’ approximation.
For a pixel in the scene observed by the camera, we canwrite the radiative transfer equation of a ray along the line ofsight to give an at-pupil radiative flux M of

4. Gas imaging detection modelˆ λ2 [

Infrared gas imaging uses radiance changes in a scene tolook for absorption or emission signatures generated by gasesMw=

ϵf (λ) B(Tf , λ) + τ f (λ) ϵg (λ) B(Tg , λ)λ1

7

N Hagen]+τ f (λ) τ g (λ) Lb (λ) dλ[≈ ∆λw ϵf,w Bw (Tf ) + τf,w ϵg,w Bw (Tg )]+τf,w τg,w Lb,w .

spectral radiance (W m 2 sr 1 um 1)

J. Opt. 22 (2020) 103001(3)

where λ1 and λ2 give the passband of the wth spectral channel, and a w subscript indicates an average across the passband. Also, B(T) is the Planck blackbody spectral radiancefunction for temperature T, τ is the spectral transmission ofa layer, and ε the spectral emissivity of a layer. For a broadband imager, λ1 and λ2 are given by the spectral range of the
optical filter on the camera. Using Kirchhoff’s law [62], we
can equate the emissivity spectrum with the absorption spectrum, ε = α and further relate the absorption and transmission
spectra, α = 1 − τ . Thus, using (3), we can model the radiancemeasured with (M′ ) and without (M) gas present as[]
Mw = ∆λw (1 − τf,w ) Bw (Tf ) + τf,w Lb,w ,[M′w = ∆λw (1 − τf,w ) Bw (Tf ) + αf,w τg,w Bw (Tg )]+τf,w τg,w Lb,w .
The difference ∆Mw = M′w − Mw gives[]∆Mw = τf,w αg,w ∆λw Bw (Tg ) − Lb,w .

M′w − Mw[].
τf,w ∆λw Bw (Tg ) − Lb,w

10measurement
spectrum, M(λ)

8

B(Tb =30°C)

6

gas absorption
spectrum, α(λ)

4
2
0
9

10
11
wavelength (um)1213density ζ = 5500 ppm.m), showing a simulated measurement
spectrum M′ (λ) (black curve) when the background source is a
blackbody at Tb = 50◦ C (red curve) and the gas temperature is at
Tg = 30◦ C. The absorption spectrum (green curve) is given at anarbitrary scale, for a thin gas cloud.

insofar as it reduces the measurement SNR. All of the variables on the right hand side of this equation are quantities that
we can measure or estimate. Equation (7) gives the gas absorption separately for each spectral channel in the system, fromwhich we can calculate its correlation with the known spectral
shape of various library gases. Together with an estimate of
the noise, this correlation allows us to calculate the probability
that the measured spectrum indicates presence of the gas. If theprobability exceeds a selected threshold, the pixel is labelled
as a probable detection. The detection at this step is labelled
as only probable because a more sophisticated algorithm cannext take advantage of spatio-temporal correlations in the gas
motion to improve the probability estimate. For example, if
gas is detected at a given pixel, then it is likely to be detected
in its neighbouring pixels as well, and the same gas parcel islikely to be found nearby in the following data frame as well.
Incorporating these spatio-temporal features, and knowledgeof gas fluid dynamics, into the detection algorithm produces
an iterative procedure that improves the detection sensitivity.
Discussions of (7) typically refer to gas absorption, implying that the gas is cooler than the background source. However,
an important case where observation of gas is seen in emission is that of a sky background. Not only does the sky occupy
a large portion of the field of view in many situations, but it
can also have a large radiance contrast. A clear blue sky background can provide over 50◦ C of thermal contrast for many
spectral bands across the 8–12 µm spectral range, if the local
humidity is low [63].

(4)

(5)

It is common to assume that the temperature of the gas andthe ambient air are approximately the same, Tg ≈ Tf , under
the assumption that the gas quickly entrains into the local air.
Inserting this into (6) and substituting the result into (5) gives
M′w − Mw].
Bw (Tg ) ∆λw − Mw

B(Tg =50°C)

12

Figure 6. An example measurement for ammonia gas (column

The denominator here is a difference of radiances (the ‘radiance contrast’) ∆L = Bw (Tg ) − Lb,w that is closely related tothe thermal contrast ∆T = Tb − Tg . The sign of the denominator indicates whether the cloud is observed in emission(∆L > 0) or absorption (∆L < 0), but the estimated αg using(5) is positive in both situations. Note that the denominatorcan come close to zero, so that the estimate for αg should be
regularized, for example by adding a small constant value tothe denominator, or by using a Wiener filter.
If the reference measurement M contains no gas, we caneasily solve for the background radiance Lb,w in each spectral
channel:)[]1 ( Mw− 1 − τf,w Bw (Tf ) .
(6)
Lb,w =
τf,w ∆λw

αg,w = [148

Figure 6 shows αg (λ) (green curve) for ammonia gas, together
with an example M(λ) (red curve) and M′ (λ) (black curve)
for a 30◦ C gas passing in front of a background blackbody
at 50◦ C. Re-arranging (4) to solve for the gas absorption, we
obtainαg,w =

16

4.1. Background-foreground separation methods

The most difficult step in equations (4)–(7) above is to determine the spectrum of the background layer M(λ) — a variantof the well-known ‘background-foreground separation problem’ from the image processing literature [64]. Perhaps thesimplest approach for separating background M from foreground ∆M begins by assuming that in a given video sequence
the background is present much more often than the foreground at any given pixel. If we simply take a running mean

(7)

Note that τf,w has disappeared from the expression: the foreground layer plays no role in the absorption estimate, except
8

J. Opt. 22 (2020) 103001

N Hagenor running median image, this mean or median can thenbe used as a rough estimate of the background. The foreground (the gas cloud, in our case) is then obtained by taking
the difference between the current image and the estimatedbackground.
A more sophisticated approach may use an adaptive Gaussian mixture model. In this algorithm, at each frame ofa video sequence, the spectrum at each pixel is modelled
as belonging to either of two Gaussian distributions, representing the background spectrum and gas spectrum. As
more frames follow, the estimates of the two distributions
improves, with the two means representing the backgroundand foreground spectra at that pixel. This approach can be
configured as an adaptive online algorithm that is not computationally expensive, so that the calculation can run inreal-time [65].
A third approach assumes that the background is a greybody, so that its spectrum follows a set of easily parametrized
curves [66]. Once we fit the data to the nearest greybody curve,
we can take the remainder of the spectrum to represent the gas
layer. Since there are many situations in which these assumptions do not hold well, researchers have also tried to relax
the greybody requirement and instead assume that the background radiance Lb (λ) is smooth: Lb (λ) varies more slowly
with respect to λ than the gas absorption cross-section spectrum σ(λ) [67, 68]. Making use of this assumption, however,generally requires a significant spectral resolution, making
measurement times long. It is also more effective on gases thathave narrow spectral features, such as methane and ammonia,
than those with broad spectral features, such as propane andpropylene [69].
A fourth approach to the separation problem is to decompose the spatial-spectral scene into its principal components,
so that if gas pixels comprise only a small number of pixels inthe scene, then the primary principal components will be unaffected by gas spectra [70, 71]. Because of the complexity of thedecomposition, however, it is hard to predict the reliability of
the algorithm for any given scene, and spatially complex structures can cause the algorithm to become unstable. Moreover,scenes in which gas clouds comprise a significant fraction of
the image make the algorithm insensitive to gas [72]. This isa dangerous property for an algorithm to have, as it makes
the system blind during the most critical situation. In addition,
the principal components approach involves a heavy computational load that can cause difficulties for video-rate spectral
imaging.

(Wavelength dependence in each parameter is left implicit.)
If the ratios of the various constituent species within the gas
mixture are known and remain constant—a common situation
for industrial sites and if the gases are of similar densities—
then the gas mixture can be characterised with single effective cross section: αmix = σ mix ρmix ℓ. Thus, while one may use
spectral unmixing [74] and matched filtering [75] algorithms
to estimate the column density images of multiple gases simultaneously, this simple solution is sometimes all that is needed.
Another complication of measuring in industrial facilities is the ubiquitous presence of steam and sometimes dust.
Although these are aerosols, both of these act as interference gases that change the measured spectral radiance, butwhich are of no interest to the final measurement. As aerosols,
these two interferents also change spectral shape dependingon particle diameters [76, 77], so that their spectral profiles
belong to families of shapes rather than a single fixed spectrum [78]. If any member of these continua of shapes has ahigh correlation with one of an instrument’s target gases, thendiscrimination between them becomes difficult using infraredalone. Instruments that employ visible cameras registered tothe same scene as the infrared images, however, have a convenient means of separating the two: steam and dust plumes
are visible, whereas most target gases are invisible.
5. Gas image column density quantification
Once gas is detected within a pixel, estimating its column
density ζ is straightforward. Equation (4) relates the absorption of the gas layer to the measured changes in radiance at thesensor. Using the Beer–Lambert–Bouguer law, the transmission is related to the concentration ρ and path length ℓ throughthe gas cloud as
ˆ ℓ
[
]
[
]
τ g (λ) = exp − σ(λ)
ρ(z) dz ≈ exp − σρ(λ) ℓ . (9)0The absorption cross-section σ(λ) (units of m2 ) is known a priori from the type of gas detected, and can be measured directlyor obtained from a spectral library, such as the NIST Infrared
Database or the PNNL infrared spectral library [79, 80].
We use ρ(z) for the gas concentration (number density) along
the line of sight and ρ (with no explicit z-dependence) as theaverage concentration along the line of sight. Because ρ andℓ cannot be separated from one another, they are grouped
together into a single quantity—the column density ζ = ρℓ—the typical units for which are ppm.m (parts-per-million times
meter).
When the gas concentration is low (the thin gas approximation), equation (9) can be written as αg ≈ σζ, and so the gas
column density becomes a linear function of the absorption.
Going from the estimated absorption to estimated gas column
density therefore requires only scaling by the cross-section:

4.2. Gas mixtures and interferents

In the presence of multiple gas species, each of which is
weakly absorbing, the overall absorption can be written as alinear sum over the component absorbances of each species[73]. Thus, for N gases, in the thin gas and homogeneous layer
approximations,ζ = αg /σ .

N
N
[ ∑
(ˆ ℓ
)] ∑
αmix = 1 − exp −σi
ρi (z) dz ≈σi ρi ℓ . (8)
i=1

0

(10)

The column density has SI units of m−2 . We can convert thisto units of ppm.m by multiplying by the density of molecules

i=19

J. Opt. 22 (2020) 103001

N Hagenunits instead of concentration. From the total volume, we geta view of the overall scale of the leak, which helps to decide
whether to issue a safety alarm. This can be something as
simple as logging a measurement when the volume of gas issmall, issuing a warning when the gas cloud is large, and causing an alert when the cloud has reached a dangerous size.

gas cloud= no gas detected= gas detection pixelfcameradetector
array

z

Py

py objective
lens

px

5.2. Estimating the volumetric gas concentration

From a safety perspective, it is important to know the volumetric concentration ρ(x, y, z) of a flammable gas cloud, as
it indicates whether the gas is capable of igniting. Methane,for example, ignites if in concentrations of between 5% and
15% (the lower and upper explosion limits, LEL and UEL),
when mixing in air at STP [82, 83]. A cloud of methane containing concentrations everywhere below the lower limit will
not ignite, even when exposed to an ignition source such as aspark or flame. Clouds of methane gas that are too rich (above
the upper limit of 15%) also do not ignite. However, highconcentration clouds quickly diffuse into surrounding air, sothat their outer edges contain a flammable border region. Thus,
high-concentration plumes are a safety hazard.
For estimating concentration, point sensors have an advantage in that they measure concentration directly, but have thedisadvantage that they can only sample the concentration at
one point. While a point-sensor measurement giving a value
above the lower explosion limit directly indicates a hazard, avalue below the LEL is ambiguous. Without a physical modelfor the gas cloud, one cannot know whether the measurement is low because the overall cloud concentration is low or
because the leak source is too far from the sensor.
Although gas imagers detect in units of column density ζrather than concentration ρ, they can obtain rough estimatesof the concentration distribution within the gas in real-timeusing heuristics, as we describe below. Imagers can also form
accurate estimates using fluid-dynamical models of the detected gas cloud [84, 85], or by simultaneously collecting projections through the cloud from multiple points of view [86, 87],
but these methods are typically too computationally intensive
to run in real-time. In practice, a rough estimate (via heuristics) can be useful as a quick initial guess for the concentration
distribution ρ(x, y, z, t).
The first step in the heuristic method is to model the gas
cloud measurement as the projection of a cylindrical shapewith its axis oriented in the direction of the gas flow, as shownin figure 8. If the axis of the gas plume motion is purely transverse (does not have a component along the view direction,z), then the estimated depth dimension of the plume becomes
the same as its height dimension. For example, in figure 9, thetransverse gas motion allows us to say that, at the position in
the plume indicated by a red border, where the height of thegas cloud is 0.52 m (23 pixels in the image), we can estimate
the depth of the plume to be approximately 0.52 m.
The second step involves a weighted backprojection of the2D data into the 3D volume, using a Gaussian-shaped weighting function such that the center of the cloud has a higher
estimated concentration than the edges of the cloud. Figure 8
shows the projection of a 1D slice from the data into a 2D

Px

Figure 7. A detector pixel projected to the location of a gas cloud.

in air (ρair = 2.687 × 10−25 m−3 at STP), and scaling the result
by 10−6 [81]. An example of applying this estimation to eachpixel in an image is shown in figure 1 for a controlled methanegas leak.
5.1. Estimating the total quantity of gas in a plume

From an image of the gas column density, we can also estimate
the total amount of gas in a cloud. That is, if we scale each pixel
to the projected area Aproj at the location of the gas cloud (figure 7), then multiply the projected area by the detected column
density, we obtain the total volume of pure gas in the pixel(units of L, m3 , or ft3 ):Vgas = Aproj ζ .
The projected area of a pixel at the gas cloud is easily
obtained by multiplying the known dimensions of the detector
array pixels (px , py ) by the image magnification, given by thelens focal length f and the distance z to the gas cloud:()()
Aproj = px z/f py z/f .
(11)
If we wish to estimate the mass of gas within the cloud, we canfirst calculate the total number of molecules N in the cloud by
multiplying Vgas by the number of molecules per cubic meterρair at the ambient atmospheric temperature and pressure. The
total mass of the gas cloud (g, kg, or lbs) is then obtained bymultiplying N by the gas molecular weight.
While the pixel size and the lens focal length are known as aresult of the hardware design, the gas cloud distance z from thecamera varies and must be estimated. One method for doing
this is to establish distances when setting up a gas cloud imaging camera at a facility. After fixing a camera’s position, theinstaller measures the distances from the camera to the primary
items to be monitored—items such as a collection of pipes, thewall of a gas container tank, the relief valves of a gas separator
tank, etc. Thus the distance at which leaks are likely to occur
can be established a priori. Other methods include using triangulation with multiple cameras, or using fluid dynamical models to compare estimated speeds of gas motion with measured
speeds.
This ability to directly estimate the total volume or mass ofa gas cloud is one of the advantages of measuring in ppm.m10

J. Opt. 22 (2020) 103001

N Hagen

plume

(a)

jet
ppm.m
3600
3200
2800
2400

(b)

2000
1600
1200
800

Figure 8. The backprojection of a column density gas measurement

400

into a cylindrical concentration volume. The vertical column of data
is taken from a slice through the estimated gas cloud shown infigure 9(b).

0(c)

volume in order to demonstrate the procedure clearly. One
can tune the degree to which the concentration is focused in
the center versus the periphery by adjusting the width of theGaussian function which sets the relative weights. The resulting tubular volume gives an estimate for the concentrations
within the cloud, where integrating through the volume reproduces the column density image.
Because the concentration estimates using the above
approach employ simple heuristics rather than a physics-basedmodel, the results give only a rough estimate of the concentration. However, it has the important advantage that it canoperate in real time. For a more accurate model, it is necessary to build a 3D fluid-dynamical model that is constrained
to give the same projection through the cloud as was measured by the camera. Doing this accurately under the weak constraint of a single measured image is difficult, but the model
can be improved through the use of multiple frames in a video
sequence. Getting this to work well is an ongoing subject of
research [87].

Figure 9. An example gas flow measurement for methane released

from a pipe through a gas flow controller: (a) a simplified physicalmodel for the jet and plume components in the flow; (b) a snapshotof one frame in the gas column density video; (c) the corresponding
gas velocity vectors. The red and blue squares in (c) indicate two
example locations for placing a boundary used to evaluate the flow
rate. The red rectangle shown in (b) indicates the region used infigure 8 as an example concentration backprojection.

When using a pixel-by-pixel estimate of the flow rate, one
way of verifying that the algorithm is behaving consistently is
to use nested boundaries, as shown in figure 9(c). Since eachboundary contains the leak source, each should produce thesame time-averaged leak rate.
While we focus on imaging methods for estimating the flow
rate, point sensor networks employing Gaussian plume modelsof the leak can also be used for estimating flow rate [29]. However, these have only been demonstrated as useful for large
leaks and smooth flow conditions. In order to expand the range
of application, Safitri et al demonstrated that one can combine
the Gaussian plume modelling with single-frame infrared gas
imaging to quantify leak rate [90]. While this allows improved
modelling of smaller leaks, it still overconstrains the profileand dynamics of the gas, affecting accuracy. Although it has
not yet been demonstrated, video gas imaging should be able
to significantly improve on this. By averaging over a long timesequence of gas detection frames, the data comes closer to thetime-average approximation required for Gaussian diffusion.
In the discussion below, we introduce the conventional
optical flow algorithm and then show why it is inadequate formost gas imaging data. Fixing the algorithm requires using
a more sophisticated implementation that constrains the flow
estimate with a fluid-dynamical model.

6. Estimating the volumetric flow rate of a leak
One of the primary advantages of a video-rate imager used forgas sensing is that its sampling is complete in the 3D (x, y, t)
space of a video sequence. Thus, pixels in one frame are neighbours of the same pixels in the subsequent frame of the video,
so that it becomes possible to design algorithms analysing
spatio-temporal correlations within the gas cloud. If a gas isfound in one area within the scene, there will be a high probability of finding it in the sample place, or close nearby, in the following frame of the video. Thus, temporal correlations can beused to improve the sensitivity of the gas detection algorithm.
Another application for using these spatio-temporal correlations is estimation of the volumetric flow rate—the gas flux
or leak rate. Existing algorithms for doing this all incorporate some variant of optical flow techniques to estimate howeach parcel of gas moves within the image. That is, multiplying the volume of a parcel of gas times its average rate of flow
away from the leak source, and summing over all parcels ofgas detected, gives an estimate of the total flux [50, 88, 89].
11

J. Opt. 22 (2020) 103001

N Hagenthe optical flow algorithm is the fluid-concentration-weighted
average velocity û along the line of sight [92–95], as in figure 10, then we can simply multiply û with the total volume Vof fluid within a pixel to give a point-by-point estimate of thefluid flux ϕ = ûV. Summing this over all pixels along a boundary gives the total amount of fluid flow across it,∑[]û(x, y)·s V(x, y) ,Φ=x,ywhere s is the unit vector normal to the boundary, so thatû·s gives the projection of the gas velocity vector across theboundary. If the boundary is closed (such as the red or blue
boundary shown in figure 9(c)) then Φ gives the estimatedoverall source flux (leak rate), which is often expressed inunits of liters per minute (lpm) or grams per second (gps). An
example measurement is shown in figure 11.

Figure 10. (a) A camera measuring the light coming from a fluid

distribution (e.g. gas cloud). (b) The integral along a line through a
fluid of concentration ρ gives a value at the image in column density
ζ. The average velocity û along the line of sight is derived from aweighted sum of the individual velocities ui .

6.2. Adapting optical flow algorithms to volumetric flow

In principle, adapting the optical flow algorithms originallydeveloped for opaque objects (i.e. (16)) to measuring volumetric fluids seems straightforward: we simply work with ζ
instead of I, and multiply each pixel’s column density by the
local velocity estimate to obtain the measured flow rate. This is
close enough to the truth as to be confusing—a problem which
we will see is further reinforced by an unfortunate choice of
nomenclature.
The established method for deriving a representation ofprojected fluid volumes for optical flow algorithms starts withthe continuity equation of fluid dynamics, which establishes
the conservation of mass [92, 93, 95]:

6.1. Principles of optical flow algorithms

Optical flow algorithms are based on the concept that when an
object moves across the image, its brightness I at each pointstays the same. This is often referred to as the ‘brightness constancy’ assumption [91]. In the context of measuring gases, the‘brightness’ represents the gas concentration, in which case it
is more appropriate to call this assumption the continuity condition, as it enforces the conservation of mass within a local
neighbourhood. The continuity condition allows us to take thebrightness within one image frame, I(x, y, t), and track its 2D
displacement (u, v) in the next frame by searching for the pathsatisfying the continuity equation
I(x, y, t) = I(x + u ∆t, y + v ∆t, t + ∆t) .

∇(ρu) + ∂ρ/∂t = u·(∇ρ) + ρ(∇·u) + ∂ρ/∂t = 0 .

This closely resembles the gradient constraint equation (16),but with the addition of ρ(∇·u) — the ‘divergence term’.
Integrating (17) along the line of sight, the continuity equation becomesˆ [∂ρ ]u·(∇ρ) + ρ(∇·u) +dz = 0 .
(18)
∂t

(12)

This path satisfying continuity givesd
I(x(t), y(t), t) = 0 ,
dt

(13)

which by the chain rule producesdI(x(t), y(t), t) =
dt

0 = u

∂I dx ∂I dy ∂I dt++∂x dt ∂y dt ∂t dt

∂I
∂I ∂I
+v + .
∂x∂y ∂t

If we consider the divergence term by itself for a moment,
we can writeˆˆ (∂u ∂v )+dz .
(19)
ρ(∇·u) dz = ρ
∂x ∂y

(14)

which by use of Leibniz’ rule becomesˆˆ z2[ ] ∂z2 (x, y)d z2∂
ρu dz =(ρu) dz + ρu z
2
dx z1
∂x
z1 ∂x
[ ] ∂z1 (x, y)− ρu z
1
∂x
ˆ z2
ˆ (
∂
∂u
∂ρ )
=
(ρu) dz =
ρ +u
dz .
∂x
∂x
z1 ∂x

(15)

In vector notation, this isu·(∇ρ) + ∂ρ/∂t = 0 .

(17)

(16)

This expression is commonly called the gradient constraint
equation, giving the relationship of the displacement vector(u, v) to the spatial and temporal derivatives of the brightness(or gas concentration).
When imaging volumetric fluids, instead of I what we actually measure is the fluid column density ζ. If the output of

Therefore, we can write this result as
ˆ
ˆ
ˆ
∂
∂ρ
∂u
ρu dz − u dz .
ρ dz =∂x∂x∂x
12

J. Opt. 22 (2020) 103001

N HagenFigure 11. Example leak rate measurement. (a) A snapshot of one frame from a 10 s long gas detection video sequence, with the gas column

density colormap superimposed on a infrared thermal image. The concentric squares indicate possible boundaries at which to evaluate thegas flow. (b) The pixel-by-pixel estimate of the flow velocities (shown with green arrows) superimposed on an image showing the gas
column density. (c) The time-resolved leak-rate estimate for the video sequence. The horizontal dashed line indicates the overall mean flux
17.2 L/min, while the black circle indicates the time location of the snapshots shown in (a) and (b).

A similar result follows for the vertical component of the velocity, v. Substituting both expressions into (19) givesˆ (ˆˆ∂u ∂v )
∂
∂
ρ
+
dz =
ρu dz +
ρv dz
∂x ∂y
∂x
∂y
ˆ
ˆ
∂ρ
∂ρ
− u dz − v dz
∂x
∂y
ˆ
( )
( )
= ∇ ûζ − u· ∇ρ dz ,(20)large pressure gradients. For incompressible flow the density
remains constant within a parcel of fluid that moves with theflow velocity. The limit between these two regimes for air isgenerally considered to be the point at which the ratio of theflow velocity to the speed of sound in air (the Mach number) ismuch less than 1. This limit almost always holds for the caseof imaging gas leaks, so that for gas imaging we are clearly in
the incompressible flow regime.
Section 6.2 showed that once we include the divergenceterm in the continuity equation, as in (22), we have a method
for analyzing volumetric fluid flow that has been shown to
work well and provide accurate quantitative results [84, 96].
However, this has produced confusion among algorithm users,
since it is also well-known that the divergence term is negligible in the incompressible regime, and these gas flows areclearly incompressible. The difficulty arises from the fact that
we are viewing the mixing flow of two different fluids, one ofwhich is transparent. Whereas ‘incompressibility’ refers to theconstraint

where û is the concentration-weighted-average velocity,´ρu dzû = ´.
(21)ρ dz
Inserting (20) into (18), we find that the first term of (18) cancels out, leaving the ‘integrated continuity equation’( ) ∂ζ∇ ûζ +
= 0.
∂t

(22)

Thus, the continuity equation (17) and not (16) should be usedas the basis for estimating the concentration-weighted-averagevelocity û.

ρgas + ρair = constant ,the gas imaging data contains only ρgas . Satisfying incompressibility for the target gas alone would mean that the concentration of a gas parcel does not change as its moves alongwith the flow. However, since the gas diffuses into the surrounding air as it propagates, the gas concentration is clearly
not constant, and so the compressible-flow regime is the appropriate choice.

6.3. Compressibility vs. incompressibility

Fluid dynamics often separates fluid flow into two regimes:
compressible and incompressible flow. For gases, the compressible regime generally holds for high-speed motion and13

J. Opt. 22 (2020) 103001

N HagenIn order to understand the impact of ignoring this distinction, we can re-derive the expression for the weighted-averagevelocity, but this time hold the divergence term to be negligible, as shown in A. What we find is that whereas (22)(which keeps the divergence term) produces velocity estimates correctly weighted by the fluid concentration along theline of sight, the no-divergence form produces velocity estimates weighted instead by the gradient of the fluid concentration along the line of sight. That is, in a naïve implementation of optical flow that ignores the divergence term (13), anylayers having larger spatial gradients, even if they have lower
amounts of gas, will play a larger role in the velocity estimate.
Additionally, any layer in which the velocity gradient is zero(i.e. spatially uniform concentration) will have no influenceat all, even if it contains the majority of the gas in the image.
This is a disaster for accurate quantification, as the error differsfrom pixel to pixel, and from frame to frame, in accordancewith the characteristics of the fluid flow. Thus, it is necessary
to work with (22) rather than (13).

at the source whereas point sensors must rely on the gas reaching the location of the sensor. The concentration is much lower
at the sensor and so the corresponding sensitivities have to
be lower in order to detect the same size leak. Another complication for comparing the two is that, while the sensitivities of both sensor types also suffer with increasing distanceand wind speeds [10, 100], infrared imagers have an additional loss of sensitivity on cloudy days and during twilight or
dawn. At these times, the air and exterior surface temperatures
are closer together, reducing the thermal contrast needed forinfrared measurement. Finally, wind turbulence places a fundamental lower bound on the sensitivity of infrared gas imaging [101].
It is also important to keep in mind that the sensitivities
reported by imaging systems are typically for measurements
taken over short time intervals. For longer measurement times(such as the 10~45 s response times of many point sensors), theimaging sensitivity can be improved through time-averaging[102]. Although very little of these results have been reported
in the literature, the author’s experience has been that one cantypically improve the sensitivity by a factor of 2~10, depending on measurement conditions. More favorable conditions
such as low wind speeds allow for a larger improvement factor.
Response time effects can cause some unexpected behavior. For example, during the hundreds of quantitative experiments that the author has performed using controlled gas leaks
or bags filled with 100% gas concentrations, not once did a
wearable personal detector set off an alarm. Although the gas
concentrations were orders of magnitude beyond the sensitivity limits of these sensors, the presence of the gas was for too
short a period to allow the point sensors to respond.
In addition to time-averaging, autonomous sensors gain
another advantage by monitoring a scene repeatedly over along period: the increased chances of ‘lucky’ acquisition, in
which the conditions are just right for achieving the highest
sensitivity. This may happen when, for example, the windchanges and begins pushing the gas towards the direction of
an infrared gas camera, increasing the size of the gas cloud. As
a result, imaging detection of methane clouds has been shownfor distances of over a kilometer, despite the high attenuationof methane signals with distance due to water-vapor.
Because of the difficulties of evaluating point sensors andgas imagers using their basic measurement units, a common metric used instead is the minimum detectable leak rate
(MDLR) of a system. In order to evaluate this metric, a detection system is deployed in a realistic measurement environment and exposed to various leak rates and various measurement conditions. (Table 1 reproduces the MDLR measurements recorded in [10].) A system with a low enough MDLR
under a wide range of conditions, and a corresponding low
false positive detection rate, will be useful for autonomous
implementation. In addition to the MDLR, we can also evaluate the quantification accuracy of these systems, and this is one
area where imaging systems have shown capabilities that point
sensor networks have difficulty reaching. For example, in a gas
volume quantification test, we can fill a bag of known volume
with a gas mixture, and release the gas at once within theimager’s field of view. The imaging sensor can then estimate

7. Experimental results and comparison to theexisting state-of-the-art
Quantitative gas detection involves measurements of gas concentration (ppm units), gas column density (ppm.m), gas cloudvolume (L), or gas leak rate (L/min). Evaluating each of thesefour measurement modes involves a different type of experiment setup. Whereas evaluating the performance metrics ofa point sensor in a lab is straightforward, it can be surprisingly difficult to evaluate point sensors in practical outdoor
conditions. Typical point sensor metrics include the quantification accuracy of the gas concentration, the lower limit ofconcentration at which gas can be detected (the ‘sensitivity’),and the response time. For flammable gases such as methane and propane, sensor measurement precisions are often
reported in the range of 1~100 ppm, but calibration fluctuations mean that measurement accuracy is worse than this.
The sensitivities of commercial sensors, however, are generally 50~150 ppm, for response times of 10~45 s [97, 98].
Although lower numbers are always better for each of these
metrics, it is not a simple process to connect them to practical use metrics, such as the minimum size of a leak thata sensor can detect, or the utility for safety alarms. In thesesituations, the response time of the sensor plays a role, as
does the distance of the sensor from the leak, and local windconditions.
Evaluating gas imagers faces similar challenges. Infrared
gas imaging systems are generally stated as having sensitivities in the range of 500~5000 ppm.m for common hydrocarbon gases [49, 99]. Response times are generally on the orderof 1 s. While column density is the basic unit of measurement
for these systems, the addition of the gas cloud size as part of
the unit can make it difficult to compare these numbers directly with the sensitivities of point sensors. If we assume thatthe length through a typical gas cloud size is on the order ofa meter, then the sensitivities appear to be much worse thanpoint sensors. On the other hand, imagers detect leaks directly14

J. Opt. 22 (2020) 103001

N HagenTable 1. Laboratory measurements of the minimum detectable leakdesigners are working to fulfill these needs with ongoingdevelopment. Point sensor networks, filtered infrared cameras, and snapshot infrared spectral imagers are the three
primary detection modes that designers are competing todevelop for industrial gas detection and quantification, withthe snapshot spectral imagers being the newest among them.
While historical usage has focused primarily on detection(the binary decision of gas ‘present’ or ‘absent’) and gas
concentration measurement, recent development has expanded into new forms of quantification: gas column density, plume volume, and leak rate. We have argued abovethat imagers provide important advantages for the these newquantifiers—especially for allowing remote estimation of leakrate.
Autonomous gas detection systems have the potential toreduce the labor burden for LDAR, and to improve our ability to quantify gas emissions from industrial sites. Camerabased systems are undergoing early deployment, and showpromising benefits over the networks of point sensors thatcurrently dominate the industry. While camera-based ‘smart
LDAR’ initially had a reputation for being less sensitive
than point sensors, industrial users have become familiar
with their different strengths and weaknesses, and camera detection has achieved widespread use in LDAR. However, camera-based systems have made fewer inroads into
safety and emissions monitoring applications, due less toany fundamental weakness in their capabilities than to thecurrent immature state of algorithm development. We haveprovided some preliminary testing information in section 7,
but much more testing is needed to adequately characterize performance. As development of these new modes
matures in the coming years, we will likely see a widespreadchange in how industry approaches safety and emissions
monitoring.
The newest detection mode, snapshot spectral imaging, also provides an improved ability to discriminate among gases and to remove the effects of interferents such as steam and dust. This allows for an
improved false-detection rate that is an essential stepin the transition to making a gas sensor system truly
autonomous.
Finally, we can also keep in mind that there are ancillary benefits to using infrared cameras for detection. In
addition to gas sensing itself, infrared camera data canalso be other purposes such as intruder detection, monitoring equipment for defects or long-term maintenanceissues, or even tracking the liquid levels in large storage
tanks.

rates (MDLR) in g/hr for gas imaging based on using a filtered
MWIR infrared-camera, at various wind speeds (reported in [10]).
Analyte0 ms−1Wind speed
0.9 ms−1

2.2 ms−1

ButaneEthaneMethanePropaneEthylene
Propylene

0.72
0.97
3.96
0.76
13.9
4.37

5.8
5.2
20.8
9.8
53.7
15.6

15.9
17.8
49.0
19.1
104
59.8

Table 2. Results of methane leak rate experiments using Rebellion

Photonics’ GCI (see table 3.24 in [39]).
Viewing
distance
(m)
17
17
17
17
30
30
30
30

Actualflow rate(L/min)Measuredleak rate(L/min)

Error
(%)

1.67
10
16.7
100
1.67
10
16.7
100

1.85
8.95
17.2
81.9
1.50
11.0
16.1
84.9

10.8
10.5
3.0
18.1
10.2
10.0
3.6
15.1

the total gas volume from the gas detection video, and this result can be compared with the bag volume. While this volumeestimation capability has been demonstrated in the field, its
accuracy has not yet been published in the literature, and so it
is not clear what the current status is of this mode.
The fourth quantification mode is the gas leak rate, typicallymeasured in g/hr or L/min. Evaluating the leak rate quantification accuracy involves placing a flow-controlled leak source
into the field of view of an imaging camera (or the center ofa point sensor network). Table 2 gives a series of measurements taken with the Rebellion Photonics’ GCI camera, fortests supervised by the US Environmental Protection Agency(EPA) [39]. Representative gas leak sizes in equipment fugitive emissions are 0.038 L/min (median) or 24 L/min (mean)[103], so the test conditions of table 2 fit into this range. The
results indicate that, under good measurement conditions of
low wind, good thermal contrast, and the leak source fully inthe field of view, the leak rate can be quantified to within an
accuracy of ± 20%.
We can also see that the quantification errors in table 2 are
actually larger than the MDLR results shown in table 1. Thus,
at the lowest gas leak levels the quantification has too mucherror to be useful—while we can detect gas at the lowest levels,a higher signal-to-noise ratio in the measurements is needed inorder to provide useful quantification.

Acknowledgment
The author was formerly an employee of Rebellion Photonics, Inc. (Houston TX, USA) as a researcher developing
hardware and software for snapshot spectral imaging devices
applied to gas sensing. However, the author is no longer
affiliated with the company and declares no conflict ofinterest.

8. Conclusions
The three gas sensing applications—LDAR, safety, and emissions monitoring—each have different needs, and sensor
15

J. Opt. 22 (2020) 103001

N HagenAppendix. Derivation of the naïve continuity
equation

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∂ρ
+v
+w
+
dz .
0=
u
∂x
∂y
∂z
∂t
According to the divergence theorem, the z-component termw ∂ρ/∂z can be neglected, leavingˆ (∂ρ∂ρ ∂ρ )
0=
u
+v
+
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(23)
∂x
∂y
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The third term inside the integral is simply
ˆ
ˆ
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d
dζ
dz =
ρ dz =,∂tdt
dt
while the first term inside the integral, expressing horizontal
motion, has the formˆ∂ρ(x, y, z, t)u(x, y, z, t)
dz .
(24)
∂x
If we approximate u(x, y, z) as a series of discrete uniform layers along z, then we can write (24) as the sum∑ui (x, y, t)

[∂

i

]
∑ ∂ρi
ρi (x, y, t) ∆z ≈ ũ
∆z .
∂x
∂x

where ũ is now the weighted average velocity formed byweighting the layer velocities ui by the corresponding gradients ∂ρi /∂x. In the integral limit of infinitely thin layers,ũ becomes the average velocity weighted by the continuous
gradient ∂ρ/∂x along z:ˆ∂ρ(x, y, z, t)∂ζ(x, y, t)u(x, y, z, t)dz = ũ(x, y, t).
∂x∂t
The same analysis follows for the vertical velocity v.
Inserting (25) and the corresponding result for v into (23)recovers another form of integrated continuity equation
ũ

∂ζ
∂ζ ∂ζ
+ ṽ
+
= 0.
∂x
∂y
∂t

(26)

The difference between this result and (22) (which keeps thedivergence term) is that while (22) produces velocity estimatesweighted by the fluid concentration along the line of sight, (26)
produces velocity estimates weighted by the gradient of thefluid concentration along the line of sight.
ORCID iD
Nathan Hagen  https://orcid.org/0000-0003-1122-9081
16

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