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The recent development of Earth observation satellites with multiangular capabilities and enhanced spectral resolution has led to preliminary attempts at determining the height of atmospheric scatterers, in particular, of top-cloud heights and smoke plumes originating from forest fires. Inspired by these previous studies, the present work presents an original methodology for the determination of the three-dimensional distribution of high-contrast atmospheric aerosols using multiangular images. The method starts with the approximately known geometry of image acquisition and a set of tie points and uses a linearized and regularized functional model to obtain the position of atmospheric scatterers identified by means of a semiassisted procedure on two or more images. A subsequent application to a CHRIS/PROBA-1 scene of Mount Etna following its eruption on June 14, 2014, allows determining the volcanic plume three-dimensional structure with a precision in the 100–200 m level.

In the last two decades, Earth observation programmes have experienced a substantial development with the usage of sensors including not only increasing geometric resolution and number of radiometric bands but also multiangular capabilities. Among the missions and sensors providing multiangular images, one may find the Advanced Along-Track Scanning Radiometer (AATSR) onboard ENVISAT (active until 2012) and its predecessor ATSR-2, which provided images with 0° and +55° along-track angles in seven radiometric bands (from 550 nm to 1200 nm) to a spatial resolution of 1 km. The Multiangle Imaging Spectroradiometer (MISR) onboard TERRA/SAR is capable of registering 0°, 26.1°, 45.6°, 60.0°, and 70.5° images in four spectral bands of the visible and near infrared with a spatial resolution of 275 m at nadir. Further, onboard the currently most recent mission, Sentinel-3 (launched in February 2016), the sensor Sea and Land Surface Temperature Radiometer (SLSTR) is designed to collect 0° and −55° images with a spatial resolution of 300 m in the highest resolution mode. All of these sensors, however, are overcome in spatial resolution and number of radiometric bands by the Compact High Resolution Imaging Spectrometer (CHRIS) onboard ESA’s PROBA-1 mission. CHRIS acquires images to a spatial resolution of 17 m (in acquisition mode 5) or 34 m (modes 1 to 4) in a large number of bands (62 in mode 1, 18 in modes 2, 3, and 4, and 37 in mode 5) for the spectral range of 400–1050 nm within nominal along-track angles of 0°, ±36°, and ±55° (see [

The most significant benefit obtained from the use of multiangular images versus the traditional use of only nadiral images derives from a better characterization of the spectral response of the surface appearing in the scene without the use of additional prior hypothesis [

In this paper, we present a methodology for the determination of the three-dimensional structure of high-contrast aerosols in the atmosphere using multiangular images of the CHRIS/PROBA-1 sensor along with its application to the determination of the vertical distribution of Mount Etna’s volcanic plume. In the following section, we explain the data preparation and the functional and stochastic models that are input to the adjustment by the least-squares method. Then we will present and discuss the results and draw the corresponding conclusions, including the limitations of the proposed methodology and possible lines for future research.

We use the VISAT-BEAM open-source toolbox [

For every image, its metadata include an estimation of the image center time (here 5:06:42, 5:05: 55, and 5:05:07 for 0°, +36°, and +55° scenes, resp.). We use these times to interpolate in the corresponding telemetry file that can be retrieved from ESA’s REDU Center (

CHRIS images usually suffer from a misalignment problem of unknown origin, which makes the image centers differ considerably from their targeted positions [

Image and terrain coordinates of GCPs for the 55° scene.

Point number | | | Longitude (°) | Latitude (°) |
---|---|---|---|---|

| 200.5 | 224.5 | 15.024593 | 37.756615 |

| 240.5 | 218.5 | 15.038412 | 37.75333 |

| 212.5 | 154.5 | 15.039819 | 37.771614 |

| 294.5 | 270.5 | 15.089861 | 37.713634 |

| 45.0 | 254.5 | 14.919533 | 37.75655 |

| 76.5 | 342.5 | 14.934227 | 37.725693 |

GCPs over the 55° scene

Next we identified homologous points in the different images of the plume using one radiometric band, where the contrast is high (band 26, 682.2 nm). It is worth noting that other studies of volcanic plumes have used spectral bands of very similar wavelengths (e.g., [

Homologous points for the 0° scene

Homologous points for the 36° scene

Homologous points for the 55° scene

The larger limitation in identifying homologous points over the 55° image is self-evident: some of the homologous points in the two other images are out of scene now; other points were included albeit with a limited degree of certainty about their correctness, while some others were more reliable.

It has to be noted that these coordinates correspond to the plume projection onto the terrain for every image. As appearing in Figure

Geometry of projective rays for a scatterer in the atmosphere.

For the sake of simplicity, we derive now the functional model based on two images. Its generalization to three or more images, to be used later, is straightforward.

After interpolation in the telemetry file, the satellite positions for the two scene shots are assumed to be known (at least approximately) in a global geodetic system like the WGS84. A right-hand set of Cartesian coordinates is defined in this system with

and

through knowledge of the geoid undulation

For a particular scatterer in the atmosphere of unknown coordinates (

Projective rays fulfill the straight-line parametric equations,

for every image

For simplicity, we can work with the first of these equations only and denote it after a slight rearrangement as

with

Consider, first, the fact that the functional model (see (

Computing the derivatives and rearranging the equation to leave the unknowns in the left-hand side, except for the residual term, we can write

Analogous expressions can be obtained for the

with

Vectors

Rectangular matrix

The resulting Cartesian coordinates can be transformed back to geodetic coordinates by making use of the well-known formulas [

in terms of the following auxiliary variables and the curvature radius already defined in (

Now the system of equations in (

Further, the system of equations in (

We pay attention now to the computation of the different approximate values and their corresponding degrees of accuracy, which will define the figures to use in the regularization. First, we have assumed that all pixels of each image are simultaneously acquired; obviously, this is only a working assumption, especially since CHRIS/PROBA uses a nonstandard push-broom acquisition technique with a nonuniform rotation speed of the platform in order to maximize image quality. However, we can explain the validity of this simplified hypothesis as a working assumption for the final 3D reconstruction of the plume as follows. Taking into account the fact that the time difference with respect to the central time in each scene can amount to 10 s [

Another working assumption is the static character assumed for the plume during the successive image acquisitions. Obviously, the effect of the wind modifies the shape of the plume, although it can be argued that the relative displacement between points belonging to the plume will be more than one degree of magnitude less than the total wind speed. For instance, a wind speed of 10 km/h (2.8 m/s) represents some 130 m of wind displacement between two consecutive images acquired with some 47 s time separation, surely leading to a

Regarding the coordinates of a particular scatterer in the atmosphere, we can use as a first approximation their planimetric coordinates

Finally, straight-line parameters can be computed from the scene shot geometry or, more precisely, in terms of the satellite and scatterer approximate coordinates. Their values result from the order of unity and the corresponding corrections to be obtained may be of the order of 0.05.

We will therefore perform the regularization of the system of equations by adding one pseudoobservation equation in matrix

and will be accompanied by the corresponding weight in the diagonal matrix

The resulting system of equations is solved by least-squares method as

The a posteriori unit weight standard deviation is obtained [

After adjustment of the model based on two images (0° and 36°), we obtain residuals typically below 100 m with a few exceptions for points 140, 142, 144, 145, and 146 reaching 150–200 m, which can be regarded as acceptable results (they are consistent with the prior precision). The a posteriori unit weight standard deviation is 0.914, and its closeness to unity confirms the validity of the adjustment. Iteration of the computations produces significant changes in neither the residuals nor the coordinates of the scatterers.

Regarding the adjustment of the model based on three images (0°, 36°, and 55°), we start by noticing an undesired result: the a posteriori unit weight standard deviation is now 2.998, which indicates that the existing errors are 3 times, on average, the expected value. By inspecting the corresponding residuals, we find out that some of them have been affected by gross errors. They correspond to some of the doubtful observations we made over the third image (55°). Starting from the largest residual (corresponding to point 142 on image 3), we iteratively eliminate the point with the worst residual in the successive adjustments as it is customary in the data snooping procedure until all the remaining residuals are acceptable. The a posteriori unit weight standard deviation is now 1.870 and the adjustment can be accepted. We observe, however, that the final residuals are mostly below 100 m for the 0° and 36° images, whereas they have a typical size of 400–500 m for the 55° images. This leads us to conclude that the error estimation for 0° and 36° images (

The three-dimensional distribution of the particular scatterers defined in the plume contour is shown over a Google Earth view in Figure

Representation of the distribution of scatterers over a Google Earth view (© 2015 Google Inc., used with permission. Google and the Google logo are registered trademarks of Google Inc.).

As a final comment, we can recall that the best observational geometry for the determination of coordinates by intersection is the one that is made at a right angle in the unknown object [

An original methodology for the determination of the three-dimensional distribution of high-contrast atmospheric scatterers using multiangular images was derived and subsequently applied to the case of Mount Etna’s volcanic plume from high-resolution CHRIS/PROBA-1 images. The final results proved to be reliable within the 100–200 m level.

The methodology presents, however, some limitations in its different steps. Regarding data acquisition, a high degree of contrast of aerosols in the atmosphere is needed to perform the necessary identification of homologous points in the different inclination scenes. The correct identification of these homologous points is a critical factor limiting the precision of the final results. Gross error testing and identification are therefore unavoidable. Another limiting factor comes from the relatively small sample of identifiable features appearing on all the different views of the plume. These limited numbers of homologous points may suffice, however, to roughly define the three-dimensional distribution of the plume. With regard to the different simplifications in the computations, we note the assumption that the plume does not move or change its shape during the total acquisition time, as well as the simplification that for every image all pixels have been acquired in a common time. Both may lead to errors assumed to be within the indicated total error budget (100–200 m). Finally, from the algebraic point of view, we may note that the problem is ill-conditioned and requires the use of regularization. It is expected that application to other cases where more images are available (the five of them or at least images on both sides of the nadir view) produces better results than those obtained here only with 0°, +36°, and +55° images and would allow drawing conclusions on the measuring precision achievable using different sets of angles.

Further research may include application to other areas and types of dense aerosols (e.g., wildfire plumes), automated measuring of homologous points over the different images, inclusion of a more detailed characterization of the platform movement during the acquisition, and the use of robust estimation to detect incorrect measurements.

The authors declare that there are no conflicts of interest regarding the publication of this paper.