**2. Materials and Methods**

The Landsat 8 OLI is a push-broom (linear array) imaging system that collects visible, near infra-red, and short-wave infra-red spectral band imagery at 30 m multi-spectral and 15 m panchromatic ground sample distances. It collects 190 km wide image swaths from ~705 km orbital altitude [28].

The OLI focal plane layout is very well described in [28] and [29]. The OLI detectors are distributed across 14 separate FPMs, each of which covers a portion of the 15◦ OLI cross-track field of view. Adjacent FPMs are offset in the along-track direction to allow for FPM-to-FPM overlap, avoiding any gaps in the cross-track coverage. The internal layout of all 14 FPMs is the same, with alternate FPMs being rotated by 180◦ to keep the active detector areas as close together as possible. This feature has the effect of inverting the along-track order of the spectral bands in adjacent FPMs. Consequently, this has the effect of inverting the signs of the cross-correlation measurements when calculating pixels offsets between PAN and MS bands, related to the volcanic cloud velocity and parallax.

The general concept of PEM methodology is that the PAN and the MS sensors on board a satellite platform cannot occupy the same position in the focal plane of the push-broom instrument. There is a physical separation between them. This separation yields a baseline and a time lag between the PAN and MS image acquisitions. Two directions are considered: The epipolar direction (EP), i.e., the azimuth direction of the satellite or the flight direction, and the perpendicular (P2E) to the EP direction. The pixel offset between PAN and MS in the EP direction is proportional to the height of the plume plus the pixel offset contribution induced by the motion of the plume itself in between the two acquisitions. The pixel offsets in the P2E direction, also controlled by the time lag, are proportional to the plume motion only, as there is no parallax in the P2E direction by definition. In principle, the offset in the P2E direction is proportional to movements of every feature in the imaged scene (e.g., meteorological clouds, lahars, rivers flow, ocean waves, vehicles). In our case study, we are interested in the volcanic cloud motion only. We use this latter information to compensate for the apparent parallax recorded in the EP offset.

If the data are downloaded in a staggered and orthorectified geometry, the processing is not straightforward, since FPMs are rotated 180◦, and the offset analysis by cross-correlation would yield opposite signs at adjacent image stripes. This hampers the correct deployment of the method described in [21]. To avoid this inconvenience, we propose the following 5 step procedure:


$$|O\_{\hbar}| = |O\_{\hbar}| - \left| O\_{p2\varepsilon} \right| \left| \tan \theta \right| \tag{1}$$

if theta is between zero and 180, or

$$|O\_{\hbar}| = |O\_{\hbar}| + \left| O\_{p2\varepsilon} \right| \left| \tan \theta \right| \tag{2}$$

if theta is between 180 and 360. *Oh* is then converted into a VCTH using the formula provided in [20]:

$$h = |O\_{\hbar}| \frac{s.H}{V.t} \tag{3}$$

for every pixel, which makes it a PEM. *h* is the plume height (m), *s* is the pixel size (m), *V* is the platform velocity (m/s), *t* is the temporal lag between the two Landsat 8 bands (s) and *H* is the platform height (m). Peculiar cases are: θ = 0◦ and θ = 180◦. In these cases, the system is no longer sensitive to plume velocity. Therefore,

$$|O\_{\hbar}| = |O\_{\hbar}|\tag{4}$$

V. Finally, the results are re-rotated to their original position. Then, one has to choose a known reference altitude value on land and attribute it to the corresponding pixel. In our case study, we choose to set to zero the coastline close to the city of Catania.

**Figure 2.** Angles and geometrical conventions, imagining this is a rotated Landsat 8 scene with a volcano at its centre. The epipolar (flight motion) direction and the perpendicular (p2e) to the epipolar direction (EP) direction are indicated. Dark gray and light gray colours indicate quadrants where the offsets are either summed up, either subtracted respectively. Ovals represent possible ash clouds directions. The light blue colour indicate the ash cloud direction of the case study presented here.

Figure 3 shows the offsets results. We show the raw results of the correlator on the left sides and the corrected results on the right side. The vertical stripes on the left sides are due to volcanic (and non-volcanic) cloud velocities and parallax: As the FPMs are inverted 180◦, the correlator yields velocities with sign opposition (as explained in the introduction). The correlation results are corrected by using |*Oe*|, *Op*<sup>2</sup>*<sup>e</sup>* as described in the above paragraphs. The pixel offsets are expressed in meters. It is interesting to note that the P2E offset correspond to cloud (volcanic cloud and non-volcanic cloud) velocities, which values are comparable to the wind speed represented in Figure 6.

**Figure 3.** (**a**): Results from the correlator show vertical stripes due to volcanic (and non-volcanic) cloud velocities. As focal plane modules (FPMs) are inverted 180◦, the correlator yields velocities with sign opposition (see text for more details). (**b**): Correlation results are corrected by using |*Oe*|, *Op*<sup>2</sup>*<sup>e</sup>* as described in this study. EP offsets (*Oe*), P2E offsets (*Op2e*). The pixel offset is expressed in meters. It is interesting to note that the P2E offset correspond to cloud velocities.

#### **3. Results and Cross-Comparisons**

Figure 4 shows the VCTH map obtained from the PEM procedure applied to the Landsat orthorectified data. The volcanic cloud height vary from about 6 up to 9.5 km above the sea level (a.s.l.) with the higher values that lie in the central region of the cloud.

The estimated VCTH has been compared the VCTH extracted by using different procedure applied to other satellite sensors. Note that the cross-comparison here is not used as a validation. It allows us to assess the consistency of the results (i.e., our results are in the same order/scale as independent measurements). A thorough validation is not possible since acquisition times (repeat cycle) of different sensors are not the same as Landsat. Since the VCTH evolves with time, we prefer to call it « cross-comparisons » rather than « validation ». In addition, different sensors acquire data from different positions, introducing a bias in the eventual validation campaign.

Here the VCTH of the Etna 26 October 2013 eruption, used for the cross-comparison with Landsat results, are estimated by using geostationary (SEVIRI) and polar (MODIS) satellite sensors.

**Figure 4.** The volcanic cloud elevation (km), extracted from Landsat 8 orthorectified image.
