Introduction
Studying and monitoring gas emissions is highly desirable since the
emitted gases can have substantial environmental impacts. This includes both
natural and anthropogenic sources such as volcanoes, industrial areas, power
plants, urban emissions or wildfires. The measurements can help to better
assess regional and global impacts of the emissions, for instance, related to
air-quality standards and pollution monitoring or climate impacts (e.g.
, , ).
Sulfur dioxide (SO2), in particular, is a toxic gas emitted both by
anthropogenic and natural sources (e.g. power plants, ships, volcanoes). The
pollutant has various impacts, both of socio-environmental and economic
nature (e.g. human health, agriculture) and on the climate (e.g. being a
precursor of stratospheric sulfur aerosols, ).
Furthermore, SO2 is an important monitoring parameter related to
volcanic risk assessment (e.g. ,
).
Passive remote sensing techniques are commonly used to monitor gas
emissions from localised emitters (or point sources). The instruments are
based on the principle of light absorption and typically measure path-integrated concentrations (column densities, CDs) of the gases.
Instrumentation can be ground, airborne and satellite based and can cover
wavelengths ranging from the near ultraviolet (UV) up to thermal long-wave
infrared (LWIR), either using solar or thermal radiation as a light source.
Note that the term “point source” is not clearly defined and may, in some
cases, refer to scales of several kilometres (e.g. a whole city in the case of
space-based observations), and in other cases, to only a few metres (e.g. a
power-plant chimney for ground-based near-source measurements).
Gas emission rates (or fluxes) of the sources are typically retrieved along a
plume transect ℓ by integrating the product of the measured CDs
with the local gas velocities in the plume. The latter may be estimated using
meteorological weather data (e.g. ) or using correlation
techniques (e.g. , ) if
the measurements are performed at a moderate sampling rate (e.g.
spectroscopic instrumentation such as COSPEC or scanning DOAS instruments,
e.g. , ) and at
sufficient source distance.
A more recent measurement technique is based on camera systems which are
equipped with wavelength selective filters (e.g.
, ,
, ,
). The imaging devices can be used to create
instantaneous CD maps of the measured species (e.g. SO2, NO2) at
high spatial resolution and at sampling rates potentially down into the
sub-Hz regime (depending on the optical set-up and lighting conditions). This
allows us to study high-frequency variations in the emission signals or to
investigate individual sources separately (e.g.
, ). As a result, the
cameras are often pointed at the vicinity of the source, where the plumes can show
turbulent behaviour, mostly as a result of aerodynamic effects and buoyancy. The
resulting velocity fields often deviate significantly from the meteorological
background wind field. Luckily, the high resolution of the imaging systems
allows us to account for these spatial and temporal fluctuations by directly
measuring the projected 2-D velocity fields using optical flow (OF) algorithms
(e.g. , ,
, , ,
).
OF algorithms can detect motion at the pixel level by tracking distinct image
features in consecutive frames. In the following, the basic principles of the
OF computation are briefly introduced as well as different optimisation
strategies (see e.g. , ,
for a comprehensive introduction into the topic). OF
algorithms are based on the assumption that a certain image quantity, such as
the brightness I or the local phase ϕ, is conserved between
consecutive frames. Then, a continuity equation of the form
∂tg+f∇ijg=0
can be used to describe the apparent motion of brightness (or phase) patterns
between two frames. Here, f=[u,v]T denotes the flow vector in the
detector coordinate system i,j. g is the conserved quantity (e.g.
I,ϕ), ∇ij=[∂i,∂j]T and ∂t
denote the spatial and temporal differentiation operators.
Equation ()
is typically referred to as the optical flow constraint (OFC) equation and
can be solved numerically per image pixel, for example using a least-squares
or a total least-squares optimisation scheme. The OFC states an ill-posed
problem, as it seeks to find the two velocity components u and v from a
single constraint (i.e. I or ϕ; cf. Eq. ). This is
commonly referred to as the aperture problem and is typically
accounted for by introducing further constraints that impose spatial
coherency to the flow field. These can be subdivided into local and
global constraints or a combination of both (e.g.
). Local methods (e.g.
) apply the coherency constraint only within a
certain neighbourhood around each pixel (the size of this aperture can
usually be set by the user). Thus, for pixel positions that do not contain at
least one trackable feature within the neighbourhood specified by
the aperture size, the algorithm will fail to detect motion. We shall see
below that this can be a fundamental problem for the emission-rate analysis
using plume imagery, in the case that extended homogeneous plume regions coincide with
a retrieval transect ℓ. The problem is less pronounced for OF
algorithms using global constraints (e.g. the algorithm by
which is used in ),
which can propagate reliable motion vectors over larger image areas. However,
note that, depending on the optimisation strategy, global
regularisers are often more sensitive to noise (e.g. )
and are typically computationally more demanding (e.g. ).
Most of the modern OF algorithms include a multi-scale analysis where the
flow field is retrieved from coarse to fine features using image pyramids
combined with suitable warping techniques (e.g. ). This
can significantly increase the robustness of the results and is of particular
relevance in the case of large displacements (i.e. several image pixels, e.g.
, ).
Optical flow intercomparison benchmarks (e.g. ,
) can provide useful information with which to assess the
performance (e.g. accuracy) and applicability (e.g. computational demands,
availability of source code) of different OF algorithms. Particularly
important for the emission-rate analysis is the computational efficiency as
well as the performance within homogeneous image regions. As discussed above,
the latter may be optimised via the incorporated coherency constraints (e.g.
by increasing the local averaging neighbourhood around each pixel)
or by performing a multi-scale analysis. However, this can lead to a
significant increase in the required computation times (e.g.
) and may therefore be inapplicable, especially for
near-real time analyses.
Given these challenges, in many cases the choice of a suitable OF algorithm
will be a trade-off between computational efficiency and the performance
within homogeneous image regions. In order to rule out potential failures in
the OF retrieval, it is therefore highly desirable to assess the OF
performance before calculating the emission rates. In this paper, we propose
a new method, which analyses an OF displacement vector field (DVF)
in order to identify and correct for potentially unphysical OF
motion estimates. The correction is performed in a localised manner, within a
specific region of interest (ROI) in the images (e.g. in proximity to a plume
transect ℓ). It measures the local average velocity vector (LAVV)
within the ROI, based on distinct peaks in histograms computed from the local
DVF. The strengths of the method are (1) that it is independent of
the choice of the OF algorithm and (2) that the additional computational
demands are small compared to the OF computation time. The new method is
introduced using the Farnebäck optical flow algorithm
() which showed promising results in
and which is freely available in the OpenCV library (e.g. ).
We use two different volcanic data sets recorded at Mt Etna, Italy and
Guallatiri, Chile to show that our method can successfully detect and
correct for unphysical OF motion estimates during the emission-rate analysis.
The paper is organised as follows: Sect. starts with a short
introduction into the technique of UV SO2 cameras and the required
data analysis. Section provides information about
the two data sets (i.e. technical set-up, measurement locations), followed by
details regarding the image analysis of both data sets
(Sect. ). The proposed correction for optical-flow-based velocity retrievals is introduced in Sect. . In
Sect. the retrieved SO2 emission rates for the Etna and
Guallatiri data sets are presented and compared to results based on (1) the
uncorrected OF DVF and (2) assuming a constant global plume velocity
using the cross-correlation lag of integrated plume intersections (e.g.
). A summary and discussion is given
in Sect. , followed by our conclusions.
Methodology
UV SO2 cameras
UV SO2 cameras measure plume optical densities (ODs) in two
wavelength windows of about 10nm width using dichroic filters. The
two filters are typically centred around 310 nm (SO2 “on-band”
filter, i.e. sensitive to SO2 absorption) and, at nearby wavelengths,
around 330 nm (SO2 “off-band” filter). The latter is used for a
first-order correction of aerosol scattering in the plume (e.g.
). An apparent absorbance (AA) of SO2 can
then be calculated based on the ODs measured in both channels:
τAA=τon-τoff=lnI0Ion-lnI0Ioff.
Here, I, I0 denote the measured plume and corresponding background
intensities. Note that all quantities in Eq. () are
a function of the detector pixel position i,j (e.g.
τAA→τAA(i,j)). The calibration of the
measured AA values (i.e. conversion into SO2 column densities
SSO2(i,j)) can be performed using SO2 calibration cells
or using data from a DOAS spectrometer viewing the plume
() or a combination of both. The
SO2 emission rates are typically calculated along a suitable plume
cross section (PCS) ℓ in the SO2-CD images
SSO2(i,j) (e.g. a straight line) by performing a discrete
integration of the form:
Φ(ℓ)=f-1∑m=1MSSO2(m)⋅veff(m)⋅dpl(m)⋅Δs(m),
where m denotes interpolated image coordinates (i,j) along ℓ,
f is the camera focal length, dpl is the distance between the camera
and the plume and Δs(m) is the integration step length (for details
see ). The effective velocity
veff(m)=v‾(m)⋅n^(m)
is measured relative to the normal n^ of ℓ (i.e.
constant in the case of straight retrieval lines) using the corresponding
velocity vector v‾(m). The velocities, if retrieved from the
images, represent averages along the line of sight (LoS) of each pixel (see
e.g. for a derivation). Since the velocity components in
the LoS direction cannot be measured from the images, the measured velocities are
approximately underestimated by a factor of cos(α) (α being
the angle between plume direction and image plane). However, to first order
(and at small angles α), this cancels out since the length of the LoS
inside the plume (and thus, the measured SO2-CDs) increases by
approximately the same cos(α) factor ().
(a) Etna overview map showing position and viewing
direction of the camera (camera cfov, fov) which was located on
a rooftop in the town of Milo. Also indicated is the summit area (source)
and the plume azimuth (plume direction). (b) Example SO2-CD image
of the Etna plume including two PCS lines (orange, blue) used for emission-rate
retrievals and two corresponding offset lines (green, red) that are used for
cross-correlation-based plume velocity retrievals (cf. Appendix ).
Position and extent of the DOAS-FOV for the camera calibration are indicated by a green
spot. Note that the displayed plume image is size reduced by a factor of 2 (Gauss pyramid level 1).
Example data
The proposed method used to correct for unphysical OF velocity vectors is applied
to two volcanic data sets recorded at Mt Etna (Italy) and Guallatiri volcano
(Chile). Both data sets were recorded using a filter-wheel-based UV
SO2 camera including a DOAS spectrometer. Details about the technical
set-up for both data sets are summarised in Table .
Instrumental set-up during both campaigns.
Etna
Guallatiri
Camera
UV camera
Hamamatsu C8484-16C
Hamamatsu C8484-16C
On-band filter
Asahi UUX0310
Omega Optical, 310BP10
Off-band filter
Asahi XBPA330
Omega Optical, 325BP12
UV lens (focal length)
25 mm
50 mm
DOAS
Spectrometer
Ocean Optics USB 2000+
Avantes AvaSpec-ULS2048x64
T-stabilisation
No (ambient)
20∘
Telescope
f=100mm, (f/4)
f=100mm, (f/4)
Optical fibre
400 µm
400 µm
Etna data
Mt Etna is a stratovolcano situated in the eastern part of the island of
Sicily, Italy. We present a short UV camera data set recorded on 16 September 2015
between 07:06 and 07:22 UTC (see Table for a technical
set-up of the instruments used for the observations). The data were recorded
during a field campaign which took place about 2.5 months prior to a major
eruptive event (i.e. in early December 2015, e.g.
). The volcano showed quiescent degassing behaviour
during all days of the campaign. The measurements were performed from the
rooftop of a building located in the town of Milo, about 10.3km
from the source. An overview map is shown in Fig. .
(a) Guallatiri overview map showing position and viewing direction
of camera (camera cfov, fov), summit area (source) as well
as the plume azimuth (plume direction). (b) Example SO2-CD
image of the Guallatiri emissions including two PCS lines used to retrieve
SO2 emission rates from the central crater (orange) and from a fumarolic
field (blue) located behind the flank in the viewing direction. An additional
line (magenta) is used to estimate gas velocities using a cross-correlation
algorithm (relative to blue PCS line; cf. Appendix ).
The position and extent of the DOAS-FOV are indicated by a green spot. Note that
the displayed plume image is size reduced by a factor of 2 (Gauss pyramid level 1).
Plume conditions
During the 15 min of data, the meteorological conditions were stable, showing
a slightly convective plume of the Etna north-eastern crater (NEC) advected
downwind (into the left image half; cf. Fig. ). The
emissions of the other craters are more diffuse and could not be fully
captured since they were partly covered by the volcanic flank. Therefore, we
kept the focus on the NEC emissions, which were investigated along two example
PCS lines located at two different positions downwind of the source
(orange and blue lines in Fig. ). A video of the Etna
emissions is shown in the Supplement video no. 1.
Guallatiri data
Guallatiri (18∘25′00′′ S, 69∘5′30′′ W,
6.071 m a.s.l.) is a stratovolcano located in the Altiplano, northern Chile. The last
confirmed eruptive events date back to 1960 (). Due
to its remote location little is known about the volcano.
The presented data are part of a short field campaign between 20 and 22 November 2014.
During the 3 days, the volcano showed quiescent degassing
behaviour from the central crater and from a fumarolic field on the SW flank
of the volcano. Due to frequent cloud abundances, only a small fraction of
the acquired data was suited for the investigation of the SO2
emissions. A cloud-free time window between 14:48 and 14:59 UTC on
22 November 2014 was chosen (see Table for details about the
instrumental set-up). An overview map is shown in Fig. .
The measurements were performed at a distance of 13.3km away from
the source.
Plume conditions
Compared to Etna, the Guallatiri emissions showed rather turbulent behaviour
with strong variations in the local velocities. The central crater plume, in
particular, changed its overall direction significantly over time, which can
be seen in the Supplement video no. 2. Emission rates were retrieved along two
(connected) PCS lines in the young plume shown in
Fig. . The lines were chosen such that the emissions
from the central crater and the fumarolic field could be investigated
separately.
Data analysis
The image analysis was performed using the Python software Pyplis
(). In a first step, all images were corrected for
electronic offset and dark current followed by a first-order correction for
the signal dilution effect. The latter was applied based on
using suitable volcanic terrain features in the images to
retrieve an estimate of the atmospheric scattering extinction coefficients in
the viewing direction of the camera. The extinction coefficients were used to
correct the measured radiances of plume image pixels for the scattering
contribution. The latter were identified using an appropriate τ
threshold applied to on-band OD images.
The sky background intensities (required to for the retrieval of AA images,
Eq. ) were determined using on/off sky reference images (SRI)
recorded close in time to the plume image data. The background retrieval was
done using the background modelling methods 6 (Etna) and 4 (Guallatiri) of
the used analysis software Pyplis (; cf.
Table 2 therein). Variations in the sky background intensities and curvature
between the plume images and SRI were corrected both in the horizontal and
vertical directions using suitable gas (and cloud)-free sky reference areas in
the plume images. All AA images were corrected for cross detector variations
in the SO2 sensitivity using a correction mask calculated from cell
calibration data as outlined by . The AA images were
calibrated using plume SO2-CDs retrieved from a co-located DOAS
instrument (cf. Table ; see Sect. for details regarding the DOAS retrieval).
The position and extent of the DOAS-FOV (field of view) within the camera images are shown in
Figs. (Etna) and (Guallatiri)
and were identified using the Pearson correlation method described in
.
The gas velocities in the plume were retrieved both using the Farnebäck OF
algorithm and the cross-correlation method outlined in
. Non-physical OF motion vectors along the
emission-rate retrieval lines were identified and corrected for using the
proposed OF histogram method, which is described in Sect. .
Note that for the analysis all images were downscaled by a factor of 2 (using
a Gaussian pyramid approach).
Etna
The required plume distances for the emission-rate retrieval were derived
from the camera location and viewing direction and assuming a meteorological
wind direction of (0±20)∘ (north wind; cf.
Fig. ). The latter was estimated based on visual
observation. The camera viewing direction was retrieved using the position of
the south-eastern (SE) crater in the images. The signal dilution correction was
performed using atmospheric scattering extinction coefficients retrieved 20 min
prior to the presented observations (i.e. from one on and one
off-band image recorded at 06:45 UTC; cf. Fig. 10 in ).
During this time the camera was pointed at a lower elevation angle and the
images contained more suitable terrain features for the correction.
Extinction coefficients of ϵon=0.0743km-1 and
ϵoff=0.0654km-1 were retrieved and used
to correct plume image pixels. The latter were identified from on-band OD
images using a threshold of τon=0.05. The dilution-corrected
AA images were calibrated using the DOAS calibration curve shown in
Appendix . The linear calibration polynomial was retrieved
prior to the analysis using camera AA values that were not corrected for the
signal-dilution effect and the corresponding SO2-CDs measured with
the DOAS spectrometer (for details see Appendix ).
Guallatiri
The plume distances were retrieved per pixel column assuming a meteorological
wind direction of (320±15)∘. The latter was estimated based
on visual observation combined with a MODIS image (see Supplement) recorded
at 15:05 UTC, in which the plume was identified. The viewing direction
of the camera was retrieved based on the geographical location of the summit
area in the images.
The dilution correction was performed using scattering extinction
coefficients of ϵon=0.0855±0.0012km-1 and
ϵoff=0.0710±0.0008km-1. The latter were
retrieved between 14:48 and 14:59 UTC using images from a second UV camera,
which was equipped with a f=25mm lens (i.e. a wider FOV) and
hence contained more suitable topographic features for the retrieval. Plume
pixels for the dilution correction were identified from on-band OD images
using a threshold of τon=0.02. An example dilution-corrected
SO2-CD image is shown in Fig. . The DOAS
calibration curve is shown in Appendix .
Radiative transfer effects
Both the Etna and Guallatiri data were recorded at long distances
(>10km). Consequently, the applied dilution correction accompanies
relatively large uncertainties of statistical nature, which we estimate to
±50%, based on . Furthermore, in-plume radiative
transfer (e.g. multiple scattering due to aerosols, SO2 saturation;
see e.g. ) may have affected the results to a certain
degree. However, both plumes showed only little to no condensation. We
therefore assess the impact of aerosol multiple scattering to be negligible. In the
case of Etna, SO2 saturation around 310 nm may induce a small
systematic underestimation in the SO2 emission rates. This is due to
the comparatively large observed SO2-CDs of up to 5×1018cm-2. The impact of SO2 saturation is, however
likely compensated to a certain degree, since the DOAS SO2-CDs (used
to calibrate the camera) were retrieved at less affected wavelengths between
Δλ0≈(315–326) nm (cf.
Appendix ). The same fit interval is used in
, who performed MAX-DOAS measurements of the Etna plume under
comparable conditions. They account for SO2 saturation by using the
weak SO2 bands between
Δλ1≈(350–373) nm (see also
) and find relative deviations of about 10 % between
the two wavelength ranges and for SO2-CDs exceeding 5×1018cm-2 (i.e.
Δλ0Δλ1≈0.9 cf. Fig. A3 in
). We therefore estimate the impact of SO2
saturation to be below 20 % for our data.
Optical flow histogram analysis
We developed a method to improve OF-based gas velocity
retrievals needed for the analysis of SO2 emission rates (Eq. )
using UV camera systems. The OF analysis of an image
pair yields dense displacement vector fields (DVFs) of the observed
gas plumes. In some areas of the image, the DVF represents the
actual physical motion of gas in the plume, while other image areas may
contain unphysical motion vectors (e.g. in low-contrast plume regions; cf.
Sect. ). The proposed method aims to identify all successfully
constrained motion vectors and, from these, derives an estimate of the average
(or predominant) velocity vector in the plume. The latter is then used to
replace unphysical motion vectors in the DVF. We recommend
performing the analysis in a localised manner, within a specific
region of interest (ROI) since the velocity fields can show large
fluctuations over the entire image (e.g. change in direction or magnitude).
Figure shows an example DVF (left)
retrieved from the Etna plume including an example rectangular ROI (top). Two
further images show the corresponding OF displacement orientation angles
φ (middle) and flow vector magnitudes |f| (bottom).
Histograms M (i.e. Mφ,
M|f|) of the motion field are plotted in the right
panels and were calculated considering all image pixels
belonging to the displayed ROI. From the images and histograms, certain
characteristics become clear:
Image regions containing unphysical motion estimates are characterised
by (local) random orientation and short flow vectors (cf. sky background pixels).
These unphysical motion vectors manifest as a constant offset
in Mφ and as a peak at the lower end of M|f|.
Image regions showing reliable motion estimates, on the other hand,
are characterised by (locally) homogeneous orientation φ and
magnitudes |f| exceeding a certain minimum length |f|min.
These successfully constrained motion vectors manifest as distinct
peaks in Mφ and M|f|.
The width of these peaks can be considered a measure of the local
fluctuations or the variance of the velocities (e.g. a very narrow and
distinct peak in Mφ would indicate a highly directional movement).
(a) Example output of the Farnebäck optical flow algorithm including
a rectangular ROI. Two further images show the corresponding orientation
angles φ (b) and magnitudes |f| (c) of the DVF.
Corresponding histograms Mφ and M|f| are
plotted on the right and include all pixels in the displayed ROI.
The histograms are plotted both including (dashed lines) and excluding (red and blue
shaded areas) short flow vectors (i.e.
|f|>|f|min=1.5pix. The orientation angles are
plotted in an interval -180∘≤φ≤+180∘ where
-90∘, +90∘ correspond to -i,+i directions and
0∘ to the vertical upwards direction (-j). The DVF was
calculated using two consecutive AA images (Δt=4.0s) of the Etna
plume, recorded on 16 September 2015 at 07:14.
Based on these histogram peaks, the proposed method derives the local
predominant displacement vector (PDV) |f|‾.
A detailed mathematical description of the analysis is provided in Appendix .
In the following, the most important steps of the analysis are described.
The retrieval of the PDV starts with a peak analysis of
Mφ and investigates whether a distinct and unambiguous
peak can be identified in the histogram. If this is the case, the expectation
value for the local movement direction φμ and the angular
confidence interval Iφ are retrieved based on the position
and the width of the main peak in Mφ (using the first and
second moments of the distribution). The analysis of Mφ
involves a peak-detection routine based on a multi-Gauss parameterisation. The
latter is done to ensure that the retrieved parameters φμ and
Iφ are not falsified due to potential additional peaks in
the distribution (e.g. a cloud passing the scene, e.g. illustrated in
Fig. ). Based on the analysis of
Mφ, a second histogram M|f| is
determined, containing the displacement magnitudes |f| of all vectors
matching the angular confidence interval Iφ and exceeding
the required minimum magnitude |f|min. Also here, an
expectation value |f|μ and confidence interval
I|f| are estimated based on the first and second moments of
the histogram.
The analysis yields four parameters
pROI=(φμ,φσ,|f|μ,|f|σ)
which are used to calculate the PDV within the corresponding ROI:
f‾(ROI)=[|f|μ⋅sinφμ,|f|μ⋅cosφμ]T.
The projected plume velocity vector for the ROI can then be calculated as
v‾(ROI)=f‾(ROI)⋅dplΔpixf⋅Δt,
where f and Δpix denote lens focal length and the pixel pitch of the
detector, and dpl is the distance between the camera and the plume. Ill-constrained
motion vectors in the DVF can then be identified with a certain
confidence based on Mφ and M|f|.
In this article, the method is demonstrated using the OpenCV
() Python implementation of the Farnebäck OF algorithm
(, also used in ). It is pointed
out, though, that it can be applied to DVFs from any motion
estimation algorithm.
Applicability and uncertainties
The proposed method offers an efficient solution to identify flow vectors
containing actual gas movement and separate them from unphysical results in
the DVF. The method is based on a local statistical analysis of the
histograms Mφ and M|f|. A number of
quality criteria were defined in order to ensure a reliable retrieval of the
local displacement parameters:
A minimum fraction rmin of all pixels in the considered
ROI is required to exceed the minimum magnitude |f|min.
The latter can, for instance, be set equal to one or can be estimated based
on the flow vector magnitudes retrieved in a homogeneous image area (e.g.
randomly oriented sky background areas in Fig. ).
The same minimum fraction rmin of pixels is required to
match the angular expectation range specified by Iφ (at a
certain confidence level nσ; cf. Appendices
and ).
If additional peaks are detected in Mφ, they are
required to stay below a certain significance value S. The latter
is measured relative to the main peak based on the integral values
(cf. Appendix and Fig. ).
If any of these constraints cannot be met, the analysis is aborted. The
settings used in this study are summarised in Table .
Please note that the method cannot account for any uncertainties intrinsic to
the used OF algorithm, since these directly propagate to the derived histogram
parameters. It is therefore recommended to assess the performance of the
used OF algorithm independently and before applying the histogram correction
(see e.g. , ). The Farnebäck
algorithm used in this study showed sufficient performance both in
and in the KITTI benchmark (cf. ).
The latter find that the algorithm yields correct velocity estimates in about
50 % of all cases (approximately 1σ). Here, “correct” means that
the disparity between a retrieved flow vector endpoint and its true value
does not exceed a threshold of 5 %. We therefore assume that the majority
(i.e. ≈3σ) of all successfully constrained flow vectors lie
within a disparity radius of 15 %. Based on this, we assume an intrinsic,
conservative uncertainty of 15 % for the effective velocities (Eq. )
retrieved from successfully constrained flow vectors. Note
that this is a somewhat arbitrary choice of the intrinsic uncertainty of the
Farnebäck algorithm, solely based on the findings of .
However, we remark again that it is beyond the scope of this paper to verify
the accuracy of the Farnebäck algorithm, which we use to illustrate the
performance of our new post-analysis method. For all ill-constrained
motion vectors which are replaced by the PDV, we assume a
conservative uncertainty based on the width nσ of the histogram peaks
(cf. Appendices and ).
Finally, we point out that the proposed histogram correction does not
constitute any significant additional computational demands. For our data
(i.e. 1344×1024pix) and on an Intel i7, 2.9 GHz
machine, the required computation time for the correction is typically less
than 0.1s. In contrast, the Farnebäck OF algorithm itself
typically requires 1.5 s (same specs.) and can be considered fast in
comparison with other solutions (e.g. ).
(a) Example flow vector field (blue lines with red dots) of
the Farnebäck optical flow algorithm for the Etna plume at 07:13 UTC including
the two PCS lines (blue and orange) and the corresponding ROIs used for the histogram analysis
(semi-transparent rectangles). Middle, right: histograms of orientation angles
Mφ and vector magnitudes M|f| for both lines (bar plot),
determined using condition
no. 7 in Appendix . The
Mφ histogram (b) also includes fit results of the multi-Gauss
peak detection (thick solid lines). The retrieved histogram parameters
(φμ,|f|μ) and expectation intervals Mφ,
M|f| are indicated with solid and dashed vertical lines,
respectively. From the corrected DVF, average effective velocities of
veff=(3.9±0.5)ms-1 (orange line) and
veff=(4.4±0.8)ms-1 (blue line) were retrieved. Note
that in the left image (1) vectors shorter than 1.5 pixels are excluded, (2) the
displayed vector lengths were extended by factors of 3, and (3) only every 15th
pixel of the DVF is displayed.
Results
The new method was applied to the Etna and Guallatiri data sets introduced in
Sect. . SO2 emission rates (Eq. )
of both sources were retrieved as described in Sect.
along the corresponding PCS lines (cf. Figs. and ).
In order to assess the
performance of the proposed correction we use the following three methods to
estimate the gas velocities in the plume:
glob is based on a cross-correlation analysis at the
position of the PCS line ℓ (i.e. the estimated velocity is applied
to all pixels on ℓ and to all images of the time series).
flow_raw uses the raw output from the Farnebäck
algorithm (i.e. without correction for erroneous flow vectors).
flow_hybrid uses reliable optical flow vectors,
identifies and replaces unphysical vectors using the DVF from the histogram analysis.
Table (in Appendix )
summarises all relevant settings for the OF-based velocity retrievals. Note that the required minimum magnitude for successfully constrained motion
vectors was set per image and ROI using the lower end of
I|f| at 1σ confidence. In order to assess the
impact of unphysical motion vectors on the retrieved
SO2 emission rates, we define the ratio κ:
κ=χpix okχall,
where χall corresponds to the SO2
integrated column amount (ICA) considering all pixels on ℓ while
χpix ok corresponds to the SO2 ICA considering only
pixels showing reliable flow vectors. κ=1, for instance, means that
all motion vectors on ℓ are considered reliable. The ROIs for the
OF histogram analysis were defined for each PCS line individually (based on
the position and orientation of the line).
Etna results
The OF gas velocities in the plume were calculated from on-band OD
(τon) images, since the OF algorithm showed best performance
for the on-band OD images (based on visual inspection before the analysis).
Figure shows an example DVF of the Etna
plume and the corresponding histograms Mφ and
M|f|. Along the orange line, the OF algorithm performs
considerably well with only 7 % of the velocity vectors found
ill-constrained. If not corrected for, these unphysical motion vectors would
result in an underestimation of only 1 % in the SO2 emission rates.
For the blue line, on the contrary, a total of 45 % of the pixels on
ℓ were found unreliable. Moreover, many of these are located in
regions showing large SO2-CDs. Hence, the impact is considerably
large and, if not corrected for, would induce an underestimation of 33 % in
the SO2 emission rates.
Time series of retrieved PDV parameters
φμ and |f|μ (dashed lines) for the two Etna PCS
lines (same colours; cf. Fig. ) and corresponding
values after applying interpolation and smoothing (solid lines). The expectation
intervals Iφ and I|f| are plotted as
shaded areas.
Time series of Etna emission rates (top panel), showing
emissions of the young (a) and the aged (b) plume of the NE
crater (orange and blue) using the two PCS lines
shown in Fig. . Emission rates were retrieved using
the three different velocity retrieval methods described above. Uncertainties
(shaded areas) are only plotted for the flow_hybrid method and the
cross-correlation method (glob). Also included are time series
of effective velocities (Eq. , middle panel) and κ values (Eq. , bottom)
retrieved from the proposed histogram analysis.
Relative deviations of retrieved SO2 emission rates shown in
Fig. for the “young_plume” (a) and the “aged_plume” (b) PCS
lines using the same colour codes as in Fig. . The
ratios are plotted relative to the results of the proposed flow_hybrid
method. Results based on the cross-correlation analysis tend to be slightly larger (by
about +14%), while the uncorrected OF velocities often yield underestimated SO2 emission rates (up to 62 %).
(a) Example output of the Farnebäck optical flow algorithm
for the Guallatiri emissions at 14:48 UTC including the two example PCS lines
(blue/orange line) and the corresponding ROIs. (b, c) Histograms of
magnitudes Mφ and orientation angles M|f|
used to retrieve the expectation intervals Iφ and
I|f| and the corresponding PDV in each ROI
(cf. Eq. ). From the latter, effective
velocities of veff=(3.1±0.5)ms-1 (crater,
orange) and veff=(1.8±0.6)ms-1
(fumaroles, blue) were retrieved. Note that in the left image (1) vectors
shorter than 1.5 pixels are excluded, (2) the displayed vector lengths were
extended by a factor of 2, and (3) and only every 15th pixel of the DVF
is displayed.
Time series of retrieved PDV parameters φμ
and |f|μ (dashed lines) for the two Guallatiri PCS lines (same
colours; cf. Fig. ) and the corresponding values
after applying interpolation and smoothing (solid lines). The expectation
intervals Iφ and I|f| are plotted
as shaded areas.
Guallatiri SO2 emission rates from the summit crater (a, orange)
and the fumarolic field (b, blue) using the two PCS lines
shown in Fig. . Uncertainties (shaded areas) are only
plotted for the flow_hybrid and the glob (cross-correlation-based) velocity-retrieval methods. Also included are the corresponding effective
velocities (from the flow_hybrid method) and the OF quality factors
κ (Eq. ). The central crater emissions show only little
variability (Φ≈0.6kgs-1), while the fumarolic emissions
are characterised by a comparatively strong emission event at 14:55 UTC
showing peak emissions of 2.5kgs-1.
Relative deviations of Guallatiri emission rates shown in
Fig. . The deviations are plotted as ratios normalised
to the results from the proposed method (flow_hybrid), both for
the crater (a, no cross-correlation results available) and for the fumarolic
emissions (b). The average ratios are 1.23±0.32% (fumaroles,
glob) and 0.85±0.12 (crater, flow_raw) and 0.75±0.22%
(fumaroles, flow_raw). Again, the latter show a rather strong variability between the images.
Prior to the emission-rate analysis, the proposed histogram method was
applied to all τon images in order to retrieve time series of
the four correction parameters
p=(φμ,φσ,|f|μ,|f|σ).
Missing data points (i.e. where the required constraint parameters were not
met; cf. Sect. ) were interpolated. The results were
averaged in time using a combined median filter of width 3 (to remove
outliers) and a Gaussian filter (σ=5, to remove high-frequency
variations in the retrieved DVFs). The results of this pre-analysis
are shown in Fig. . Due to the stable
meteorological conditions the retrieved parameters show only little variation
with average values of φμ‾=(-58.3±5.1)∘
and |f|μ‾=(0.93±0.09)pixs-1
(orange line) and φμ‾=(-78.5±3.1)∘ and
|f|μ‾=(1.04±0.06)pixs-1 (blue
line).
Figure shows the results of the
SO2 emission-rate analysis for both PCS lines and the three different
velocity retrieval methods. Also included are the corresponding effective
velocities (average along ℓ, second panel) and the retrieved κ
values (Eq. ). The latter indicates the percentage impact of
unphysical OF motion vectors on the SO2 emission rates. The plotted
uncertainties in the SO2 emission rates and the effective velocities
(shaded areas) were calculated as described in Appendix .
SO2 emission rates between 4.9 and 9.7 kgs-1 (average of
7.1 kgs-1) and 4.8–10.7 kgs-1 (average of
7.8 kgs-1) were retrieved along the orange and blue lines,
respectively, using the proposed flow_hybrid method. The slightly
higher values in the aged plume are likely due to the fact that this line
captures more of the emissions from the other Etna craters (cf. Supplement
video no. 1). The corrected OF emission rates show good agreement with the
results using the cross-correlation velocities (glob method). The
latter, however, tend to be slightly increased by about +14% (cf.
Fig. ). The flow_raw method (i.e.
uncorrected OF velocities), on the contrary, often yields significantly
decreased SO2 emission rates, especially in situations where
unphysical OF motion vectors coincide with either of the retrieval lines
(i.e. low κ value; cf. Fig. ). The latter
show rather strong fluctuations between consecutive frames (i.e. local
scatter in the κ values) with an average impact of
κ‾=(0.68±0.15). These fluctuations are due to the
somewhat random nature of the initial problem. Namely, that the occurrence
(and position) of regions containing unphysical motion vectors can change
significantly between consecutive frames (cf.
Fig. ). These unphysical fluctuations
(in the estimated gas velocities) directly propagate to the
SO2 emission rates (retrieved using the flow_raw method)
and are thus not to be misinterpreted with actual (high-frequency)
variations in the SO2 emission rates.
Relative deviations of the three methods are shown in
Fig. (normalised to the results from the proposed
flow_hybrid method). The cross-correlation-based retrievals
(glob) tend to yield slightly larger SO2 emission rates (by
+14% on average), while the uncorrected OF (flow_raw) often
shows underestimated results (by -20% on average). However, we point out
again that these underestimations generally show a rather strong
variability. This includes cases showing considerably large underestimations
(up to 62 %) and other cases in which the OF algorithm appears to perform
sufficiently well (i.e. ΔΦ=1 in Fig. ).
Guallatiri results
The OF gas velocities for the Guallatiri data were retrieved using the
on-band OD images. An example DVF is shown in
Fig. . Here, the two sources are clearly
separable, showing a convective central crater plume (approx. location at
cols. i≈50–80) and the emissions from the fumarolic field
located behind the volcanic flank (i≈100–300). Also included
are the results of the proposed OF histogram analysis, which was performed
relative to the two displayed PCS lines used for the
SO2 emission-rate analysis (cf. Fig. ).
In this example, the OF algorithm performed considerably well. The
uncorrected OF would therefore result in small underestimations of 6 %
(crater) and 3 % (fumaroles) in the SO2 emission rates. The
different plume characteristics of both sources can be clearly identified
based on the displayed histogram distributions. The central crater plume
(orange) rises almost vertically
(φ=(-12.4±17.5)∘) and reaches velocity magnitudes of up
|v|max=4.5ms-1. The fumarolic emissions (blue) are less convective (φ=(+55.6±17.4)∘) and show
slightly smaller velocities with maximum magnitudes of
|v|max=3.4ms-1.
The time series of the interpolated and smoothed displacement parameters for
both PCS lines is shown in Fig. . Compared to
Etna, the two plumes show considerably more variability both in orientation
and in the velocity magnitudes (cf. Figs. and ).
The resulting average values are
φμ‾=(12.6±16.8)∘ and
|f|μ‾=(1.17±0.33)pixs-1 (crater)
and φμ‾=(15.9±13.1)∘ and
|f|μ‾=(1.30±0.13)pixs-1
(fumaroles). Due to the rather strong temporal variations, the emissions of
both sources could not always be successfully separated using the two (fixed)
PCS lines. This can be seen in the Supplement video no. 2, which shows the
evolution of SO2 emission rates for both PCS lines.
The results of the emission-rate analysis are shown in
Fig. , again, including effective velocities and
κ values for both PCS lines (cf. Fig. ). As in
the Etna example, the SO2 emission rates were calculated using the
three different velocity retrieval methods introduced above (i.e.
glob, flow_raw, flow_hybrid). In general,
similar trends can be observed. The uncorrected OF often causes significant
underestimations in the SO2 emission rates. It furthermore
accompanies rather strong (and unphysical) high-frequency fluctuations which
are propagated to the SO2 emission rates (see Sect. for a discussion). The cross-correlation velocity analysis
could only successfully be applied to the emissions from the fumarolic field
(cf. Fig. and Sect. ),
since the central crater plume showed too strong fluctuations both in space
and time. The corresponding emission rates of the fumarole emissions
(Fig. , right, purple) show good agreement with
the flow_hybrid method.
The SO2 emission rates, which were calculated based on the proposed
flow_hybrid method, show only little variation in the central
crater emissions with values ranging between 0.1 and 1.5 kgs-1
(Fig. , left). The corresponding fumarole emissions,
however, show rather strong variations with peak emission rates of
2.5kgs-1 (at 14:55 UTC), even exceeding the observed amounts from the central
crater. The sum of both sources yields total
SO2 emission rates of
Φ‾tot=1.3±0.5kgs-1 with peak
emissions of up to 2.9kgs-1.
Relative deviations of the retrieved SO2 emission rates between the
three velocity methods are shown in Fig. . As in the
case of Etna, the cross-correlation-based results (glob, fumaroles)
tend to be slightly increased (here +23%), while the uncorrected OF
(flow_raw) results in an average underestimation of -20%.