Geometric estimation of volcanic eruption column height from GOES-R near-limb imagery – Part 1: Methodology

. A geometric technique is introduced to estimate the height of volcanic eruption columns using the generally discarded near-limb portion of geostationary imagery. Such oblique observations facilitate a height-by-angle estimation method by offering close-to-orthogonal side views of eruption columns protruding from the Earth ellipsoid. Coverage is restricted to daytime point estimates in the immediate vicinity of the vent, which nevertheless can provide complementary constraints on source conditions for the modeling of near-ﬁeld plume evolution. The technique is best suited to strong eruption columns with minimal tilting in the radial direction. For weak eruptions with severely bent plumes or eruptions with expanded umbrella clouds the radial tilt/expansion has to


Full disk fixed grid image 95
We use the highest resolution ABI 0.64 µm (band 2) level 1B radiances. The full disk view, which covers the entire Earth disk, is a 21696´21696-pixel image given on a fixed grid rectified to the Geodetic Reference System 1980 (GRS80) ellipsoid. The ABI fixed grid is an angle-by-angle coordinate system that represents the vertical near-side perspective projection of the Earth disk from the vantage point of a satellite in an idealized geosynchronous orbit 35,786 km above the Equator (GOES-R PUG L1B Vol 3 Rev 2.2, 2019). The east-west and north-south fixed grid coordinates increase by exactly 100 14 µrad per pixel in the final resampled image. This 14-µrad instantaneous field of view (IFOV) corresponds to a 500-m ground-projected instantaneous field of view (GIFOV or horizontal spatial resolution) at the equatorial sub-satellite point observed with a view zenith angle (VZA) of ~0º (satellite elevation angle ~90º), as sketched in Fig. 1. At the near-limb locations of Kamchatka and the Kurils, however, the same IFOV corresponds to a GIFOV of ~4 km, because these areas are observed at grazing angles with VZA > 80º (satellite elevation angle < 10º). The ~8 times larger pixel footprint near the limb 105 renders images of horizontal surface features very blurred.
In contrast, the side of a vertically-oriented object, such as a mountain peak or eruption column, is observed at a satellite elevation angle (relative to the side) of almost 90º near the limb-the roles of zenith angle and elevation angle are reversed for vertical orientation. Thus, the local vertically-projected instantaneous field of view (VIFOV or vertical spatial resolution) is only slightly coarser than the equatorial nadir GIFOV, because it scales linearly with distance to the satellite. For example, 110 the Sheveluch volcano in northern Kamchatka is located ~14% further from GOES-17 than the sub-satellite point and observed at VZA = 83.4º, leading to a VIFOV of ~573 m. Thanks to this fine VIFOV, even small vertical features that would be sub-GIFOV were they oriented horizontally can in fact be identified in near-limb images as they protrude from the ellipsoid.
The AHI angular sampling distance in the 0.64 µm visible channel (band 3) is slightly smaller than 14 µrad, resulting in a 115 projected locations of T and the vector connecting them are known. But base point B, that is the nadir projected location of point T, is generally unknown. For the special case of a near-vertical column of ejecta, however, the base location can be approximated by the surface (x,y) coordinates of the volcanic vent, which is available from geographical databases. 160

Method 1: height from sensor-projected length
The above-ellipsoid height of column can be estimated from its sensor-projected length in the x-y plane and the view zenith angle simply as assuming a flat Earth. Sensitivity to a given error in the measured projected length decreases quickly with increasing view 165 zenith angle. However, the GIFOV also increases rapidly at large . For example, a one-pixel error in projected length at = 84° is ~4 km, which translates to a height error of 420 m. In practice, it can be difficult to accurately determine point P at very oblique view angles when using a traditional map projection due to potentially severe image distortions (e.g. eastwest stretching in equirectangular projection). For the highest plumes Earth's curvature also has to be accounted for.

Method 2: height from true shadow length 170
In the same manner as described above, column height can also be estimated from the solar-projected column length (i.e. This method yields column height above the surface on which the shadow is cast. Eq. (2) corresponds to the simplest case of a flat ocean or flat cloud surface; in the latter case, the absolute height above the ellipsoid can be obtained by adding the 175 estimated cloud height. For shadows over land the calculations are more complex and require a digital elevation model (DEM) to remove topography effects. A volcanic plume with a particularly bumpy top presents additional difficulty, because the highest point (e.g. overshooting top) might cast its shadow on the lower (and wider) parts of the plume rather than on the surface. In that case, the surface-measured shadow length leads to a height that underestimates the maximum plume height.
The height estimate is formally less sensitive to errors in measured shadow length at large solar zenith angles around 180 sunrise or sunset. Determining the end of shadow location, however, can be particularly difficult at these times if the shadow falls near the day-night terminator. Another complication is that at certain sun-view geometries, the length of the observed shadow differs from that of the true shadow-'true' (stick) shadow length is defined by Eq. (2) as height times the tangent of solar zenith angle. For the special case of a narrow and tall vertical column, the potential difference between the observed and the true shadow lengths can only be negative: when the sun and satellite are on the same side of nadir (small relative 185 azimuths), part of the true shadow might be obscured by the column itself. For most other sun-view geometries (medium to large relative azimuths), however, the entire true shadow extending from the base of the column is observed. In either case Eq. (2) can be used, because the starting point of the shadow (the vent location) is known, even when obscured, and, thus, the true shadow length can be determined if the terminus of the shadow is clearly observed.
For a horizontally extended ash layer detached from the surface, on the other hand, the error in the observed shadow 190 length can be both negative and positive, and Eq. (2) is applicable to nadir satellite views only. The case of a suspended ash layer under arbitrary viewing and illumination conditions is discussed in the next section.

Method 3: height from distance between plume edge and shadow edge
The generalization of the stick shadow method to the more common case of a horizontally expanded cloud or ash layer was derived by Simpson et al. (2000a, b) and Prata and Grant (2001). Here layer height is determined from the direction and 195 length of vector D, which connects the terminus of the shadow (S) with the sensor-projected image location (P) of the leading edge of the plume (T). Vector D in this case is the observed (apparent) shadow, whose length is generally different from the true shadow length defined by Eq.
(2), as demonstrated in Fig. 3b for the principal plane. For example, if the satellite and the sun are on the same side of the nadir line and < # (satellite above the sun, sat 1 position) the observed shadow ( 7 ) is foreshortened relative to the true shadow ( ), and if > # (satellite below the sun, sat 2 position) no 200 leading-edge shadow is observed due to obscuration by the plume. In contrast, if the satellite and the sun are on opposite sides of the nadir line (sat 3 position), the apparent shadow ( 9 ) is longer than the true shadow, because the satellite also observes shadow cast under the leading edge by other parts of the plume.
Direction G is independent of h, as it is a function solely of the sun-view geometry angles. Once the separation distance 220 | | between the plume edge and the shadow edge along azimuth G is determined, the height of the plume edge can be , where − # is the relative azimuth angle. Eq. (8) is the generalization of Eq.
(2), with the true shadow length ( ) in the numerator being replaced by the apparent shadow length and tan # in the denominator being replaced by a more 225 complicated formula, which depends on the view zenith and relative azimuth angles too.
Note that for a horizontal suspended ash layer only a single height estimate can be derived from the (edge) shadow using Eq. (8), which becomes Eq. (2) for nadir viewing. A vertical column, however, represents a special case, for which two separate height estimates can be derived from the shadow, using both Eq. (2) and Eq. (8). This is so because for a column, the surface projected location of top point T is known from three different directions: the satellite view (point P), the solar 230 view (point S, the shadow terminus), and the effective nadir view (base point B). The vector (parallax) between any two of these surface projections can be used, in conjunction with the sun-view angles, to estimate the height: method 1 for (sensor projected length), method 2 for (true shadow length), and method 3 for (the 'apparent shadow', although this line segment is not in shadow for a column). For a suspended ash layer, however, the nadir-projected base point B is unknown and thus only method 3 is applicable. 235 In the practical implementation of Eq. (8), the satellite image is first rotated so that one of its axes aligns with azimuth G and then the plume-shadow separation distance can be easily calculated along an image row or column. The plume and shadow edges can be delineated visually by a human observer or more objectively by a wide variety of edge detection algorithms (Canny, Roberts, Sobel, Prewitt, Laplacian, etc.). As before, a DEM is needed to remove topography effects when shadows over land are analyzed. 240

Method 4: height from stereoscopy
The previous three methods estimate plume height from a single satellite image. This is possible in two special cases when an extra piece of information can be recovered from the image in addition to the sensor-projected plume top location. One, for vertical eruption columns the location of the base (i.e. the nadir-projected location of the top) can be approximated by that of the volcanic vent. Two, when shadows are visible they provide the projected location of a plume top/edge point from 245 a second (the solar) perspective. Therefore, these single-image methods can be considered as effective 'stereo' methods, because they use the surface locations of a plume point projected from two different directions.
In the general case, column/plume height can be estimated by proper stereoscopy utilizing multiple views from at least two different directions (e.g. de Michele et al., 2019;Zakšek et al., 2018). Note that the formulas derived in section 3.3 can also be used as a simplified stereo algorithm, if the shadow terminus location and the solar zenith/azimuth angles are 250 replaced respectively by the sensor-projected plume location and the view zenith/azimuth angles corresponding to a second satellite. In this case D is the parallax vector between the two satellite projections. Applying the generalized shadow method in stereo mode implies the assumptions that (i) the satellite images are perfectly time synchronized and (ii) the two lookvectors, which connect the projected plume locations to the corresponding satellites, have an exact intersection point. If the images are asynchronous, plume advection between the acquisition times has to be corrected for. Furthermore, look-vectors 255 never intersect in practice due to pixel discretization and image navigation uncertainties. Therefore, dedicated stereo algorithms use vector algebra to search for the height that minimizes the distance between the passing look-vectors rather than rely on the analytical solution derived from view zenith and azimuth angles with the assumption of exact line intersection.
In Part 2, we use both the shadow-stereo method and a dedicated stereo code for validation. The analytical solution of 260 method 3 is applied to plume top features that could be visually identified in both GOES-17 and Himawari-8 images. Limb imagery is generally unsuitable for automated stereo calculations due to the difficulty of pattern matching between an extreme side view and a less oblique view. A human observer, however, can still identify the same plume top feature even in such widely different views.
We also use a novel fully-automated stereo code to retrieve plume heights of the 2019 Raikoke eruption. This "3D 265 Winds" algorithm, which was originally developed for meteorological clouds, combines geostationary imagery from GOES-16, GOES-17, or Himawari-8 with polar-orbiter imagery from the Multiangle Imaging SpectroRadiometer (MISR) or Moderate Resolution Imaging Spectroradiometer (MODIS) to derive not only plume height but also the horizontal plume advection vector (Carr et al., 2018(Carr et al., , 2019Horváth et al., 2020). The technique requires a triplet of consecutive geostationary full disk images and a single MISR or MODIS granule. Feature templates are taken from the central repetition of the 270 geostationary triplet and matched to the other two repetitions 10 min before and after, providing the primary source of plume velocity information. The geostationary feature template is then matched to the MISR or MODIS granule which is observed from a different perspective, providing the stereoscopic height information. The apparent shift in the pattern from each match, modeled pixel times, and satellite ephemerides feed the retrieval model to enable the simultaneous solution for a horizontal advection vector and its geometric height. 275

Note on potential azimuth distortions in mapped satellite images
At this point, it is worthwhile to note the possibility of angular distortions in a map projected satellite image, because this caveat is usually ignored in geometric height retrievals. Methods 1 and 2 require respectively the sensor-projected column length along the view azimuth and the shadow length along the solar azimuth # , while method 3 requires the distance between the plume edge and the shadow edge along the azimuth G . If the image is in a non-conformal map projection, 280 these angles are not preserved locally.
The equirectangular projection (Plate-Carrée), used in NASA Worldview and also implicit in the gridded CEReS AHI data, is a non-conformal projection that has a constant meridional scale factor of 1 but a parallel (zonal) scale factor that increases with latitude j as sec( ). The non-isotropic scale factor leads to considerable east-west stretching and azimuth distortion at the latitudes of Kamchatka and the Kurils. The magnitude of angular distortion depends on azimuth and can 285 easily be 10°. Angular distortion has to be considered or a locally conformal map projection needs to be used when applying methods 1-3.

Measurement principle
This method is essentially the same as method 1, but instead of calculating linear distances in a conventional map projection, 290 it determines the angular extent of an eruption column from the ABI fixed grid image. Operating in the angular space of the fixed grid has the advantages of (i) working with a more 'natural', less distorted view of a protruding column (e.g. no zonal stretching), (ii) the VIFOV being considerably smaller than the GIFOV, and (iii) no Earth curvature effects. The geostationary side view geometry of the measurement is sketched in Fig. 4. In the following we ignore atmospheric refraction effects, which will be shown to be largely negligible in section 4.2. 295 The satellite coordinate system has its origin located at the satellite's center of mass. The x axis (Sx) points from the satellite to the center of the Earth and the upward pointing z axis (Sz) is parallel to the line connecting the center of the Earth with the North Pole. The y axis (Sy) is aligned with the equatorial axis and completes the right-handed coordinate system.
The look vectors connecting the satellite to base B and to the ellipsoid projected location of top T, that is P, are respectively =`* ,b , *,c , *,d e and =`+ ,b , +,c , +,d e. For convenience, Fig. 4 depicts an eruption column on the 300 meridian of the sub-satellite point and thus plots the ellipsoidal cross section along the Sx-Sz plane. In the general case, the shown cross section corresponds to the cutting plane defined by the Sx axis and look vector , which is obtained by rotating the Sx-Sz plane around the Sx axis by the geocentric colatitude. The image location of B is determined from the known geodetic latitude and longitude of the volcano. Calculation of the satellite-to-pixel look vector and the geodetic latitude and longitude of a given ABI image pixel is described in Appendix A. 305 The angle between the top and base look vectors can be determined from the dot product of and as If is expressed in radians, an estimate of column height h can be obtained simply as This estimate is the projection of the column height to the axis k , which is perpendicular to look vector and tilted slightly 310 from the local vertical axis Z by an angle of 90°− . The projection of T onto the o -× o plane by look vector is point q . The foreshortening due to the slight rotation from the exact limb ( = 90°) can be corrected by dividing ℎ i by cos(90°− ), which is an almost trivial correction of 154 m / 10 km at = 80° and 55 m / 10 km at = 84°. Note that the angular distance between the geodetic zenith and the geocentric zenith (or the angle of the vertical) equals the difference between the geodetic latitude and the geocentric latitude, which is ~0.18º for Kamchatka. This is a negligible difference for 315 our purposes and thus axis Z can be either the geodetic or the geocentric vertical.
The horizontal expansion of the plume top in the radial direction, or equivalently the radial tilt of an eruption column with no umbrella cloud, can however introduce substantial biases. Expansion of the umbrella cloud towards the limb (away from the satellite), depicted by red in Fig. 4, leads to an overestimated angle V between the base and top look vectors and positive height bias. Conversely, plume expansion away from the limb (towards the satellite), depicted by blue in Fig. 4, 320 leads to an underestimated W and negative height bias. To minimize such biases for an expanded umbrella cloud, one has to estimate the plume point closest to the vertical at the vent and use that as point T in the calculations. If under weak winds the plume expansion is fairly isotropic and advection is small, the center of the plume-determined visually or by fitting a circle to the umbrella cloud-is a good choice for T (see the top middle inset in Fig. 4). Under strong winds, a point near the windward plume edge might be used instead. The approximate nature of locating the plume point with the smallest radial 325 distance to the vent, however, introduces an inevitable uncertainty in the height estimate.
In contrast, the tilt of the eruption column relative to o in the o -× o plane can be corrected for (see the top right inset in Fig. 4). The sideways tilt angle can be determined from the dot product of o and vector u connecting base B to q : Here ℎ yz{ = 35,786,023 m is the satellite (perspective point) height above the ellipsoid, }~= 6,378,137 m is the semi-335 major axis of the GRS80 ellipsoid, and vector connects the center of the ellipsoid to base B. The final height estimate is then obtained as ℎ iˆ= ℎ i XYZ(‰) XYZ(Š#°W/) .
In the actual implementation of the algorithm, we up-sample ABI images by a sub-pixel factor (SPF) of two using bilinear interpolation. This follows the exact practice of the operational ABI image navigation and registration assessment 340 tool, which also refines ABI images to half a pixel resolution (Tan et al., 2019). Here it is relevant to note that the native spatial sampling by the ABI detectors (10.5-12.4 µrad) is finer than the pixel resolution (14 µrad) of the level 1B product (Kalluri et al., 2018). Strictly for visual clarity, images in subsequent figures are magnified further; however, all height calculations are performed on data up-sampled with SPF = 2.

Refraction effects 345
The geometry of terrestrial refraction is sketched in Fig. 5. For spaceborne observations, the known quantity is the unrefracted view zenith angle of a pixel, which is the zenith angle at the intersection of the idealized (prolonged exoatmospheric) ray with the Earth ellipsoid (point P). An actual outgoing ray ascending through the atmosphere is refracted away from the zenith; hence, its zenith angle increases with height as it slightly bends toward the satellite sensor.
Consequently, the apparent image position of the terrestrial source of the ray is displaced in the radial direction to a point 350 with a larger (unrefracted) view zenith angle, that is, closer to the limb (i.e. from point ′to point P). A grazing ray traverses a substantial range in latitude and/or longitude and is subject to fluctuations in weather, which are complicated to handle properly. Most Earth remote sensing applications, however, can rely on the analytical treatment of refraction by Noerdlinger (1999), which was designed to derive corrections to geolocation algorithms. This method assumes a spherical Earth and spherically symmetric atmosphere and relates three different angles by simple 355 analytical formulas: the known zenith angle of the unrefracted ray at surface point P, the zenith angle i of the same unrefracted ray at the vertical of the true (refracted) Earth intersection point ′, and the zenith angle ′ of the refracted ray at surface point ′. (In the notation of Noerdlinger (1999) these angles are respectively # , , and ′.) In general, > i > ′.
For the calculation of the bidirectional reflectance distribution function, the zenith angle difference − ′ is the relevant refraction effect. For the calculation of the apparent horizontal displacement of the observed point, i.e. the distance between 360 ′ and P, however, the zenith angle difference − i is needed.
The calculations by Noerdlinger (1999) for 'white light' (0.46-0.53 µm), sea level, a surface temperature of 15°C, and a refractive index at the surface of = 1.000290 yield a linear displacement of 448 m at = 80° and a range of 1267-4858 m at = 83°-86° typical for Kamchatka and the Kurils; these numbers could vary by ~25% depending on weather. Using the Ciddor (1996) equation for the refractive index-which is the current standard and is a function of wavelength, temperature, pressure, humidity, and CO2 content-slightly reduces the surface refractive index at 0.64 µm to = 1.000277 and the original Noerdlinger (1999) displacement values by a few dozen to a few hundred meters. Considering that the band 2 GIFOV (or ground sample distance) rapidly increases near the limb and reaches ~4 km at = 84°, the horizontal displacement due to refraction could be a relatively modest 1-2-pixel shift at the surface.
More importantly, however, the horizontal displacement of the surface base point does not affect our height estimates, as 370 long as ABI pointing and angular sampling are stable, because the pixel location of B is fixed by the known geodetic latitude/longitude of the volcano, rather than by searching for the (potentially shifted) position of the volcano's feature template within the full disk image. The height calculation is only affected by a shift in the position of top T, because T has to be located in the image by visual means. The refractive displacement scales as ( − 1.0), which in turn is approximately proportional to pressure. Therefore, the linear displacement at 5 km altitude (~500 hPa) is about half of the surface value and 375 less than a third of that at the tropopause. This amounts to a practically negligible sub-pixel shift for plume tops above 5 km.
For plume tops below 5 km, one might apply a general 1-pixel radial shift away from the limb; such a first-order correction works well for the Kamchatka volcanic peaks we use in the next section to validate the height retrieval technique.
We note that refraction is not considered in the operational ABI image navigation, because its effect has been deemed marginal compared to the variation of measured geolocation errors (Tan et al., 2019). We also note for completeness that 380 refraction changes the apparent solar zenith angle at low Sun and thus also affects the shadow-based height estimation methods through tan # . In this case, the relevant quantity is the angle difference # − # • , which is ~0.19° at # = 84°, causing a relatively small ~300 m / 10 km underestimation.

Validation using volcanic peaks
Volcanic peaks in the Kurils and Kamchatka provide a large set of static targets for the validation of the side view height 385 estimation method. Figure 6 exemplifies the GOES-17 fixed grid view of central Kamchatka on two different days, when several volcanic peaks were clearly visible. The images demonstrate that viewing conditions, what astronomers call 'seeing', can vary significantly from day to day and even diurnally, because of changing turbulence, lighting, and haziness. Therefore, a given mountain peak can easily be identified on certain days but not on others. Here the base point (red diamond) was fixed by the geodetic coordinates of the vent, while the peak position (blue diamond) was visually determined and then corrected for refraction by applying a 1-pixel inward radial shift; that is, the top pixel T used in the height calculations is the one located 1 pixel 'below' the visual peak in the rotated images. Note that the distance between the base and peak pixel increases with increasing summit elevation. Figure 7c also highlights that identification of peaks is often the easiest when the volcanoes peek through a lower level cloud layer. 395 Figure 8 demonstrates the height estimation through the example of Kronotsky. Using the marked base and top pixels, the side view method yields a height estimate of 3548 m, which is in excellent agreement with the true height of 3528 m. Figure 8a shows the fixed grid view with base-relative isoheight lines drawn as visual aid. These contour lines were obtained by calculating the height between the base pixel and every single image pixel as described in section 4.1. In practice, this amounts to determining the intersection of a given pixel's look vector and the plane that contains the base point and is 400 perpendicular to the base pixel's look vector (i.e. the o -× o plane in Fig. 4). The visually identified top pixel is located correctly between the 3 km and 4 km contour lines.
The GOES-17 image in the traditional equirectangular projection is plotted in Fig. 8b. The mapped image is severely distorted compared to the natural fixed grid view, as it is stretched in the parallel (east-west) direction by a factor of sec( = 54.75°) = 1.73. This makes image interpretation and the precise identification of the peak more difficult. The non-405 isotropic scale factor also leads to considerable azimuth distortions. Although the true GOES-17 view azimuth is = 113°, the mapped image of the volcano lies along the apparent (distorted) azimuth of = 103°. The ellipsoid projected distance between the base and our best visual estimate peak location is ~31 km, which yields a height of 3768 m using Eq. (1) with a view zenith angle of = 83.07°. Although the height estimate is fairly decent in this particular case, the severe image distortions make height calculation from linear distances measured in a conventional map projection generally inferior to the 410 angular technique facilitated by the fixed grid side view.
For a more comprehensive validation, we selected 50 mountain peaks ranging in elevation from 502 m (Mashkovtsev) to 4835 m (Klyuchevskoy). The geodetic latitude/longitude and the elevation of a given peak can differ slightly between databases (Global Volcanism Program, Google Earth, KVERT list of active volcanoes), due mainly to the use of different reference ellipsoids (the exact choice of which, however, is usually undocumented). For example, the elevation of Kronotsky 415 is 3482 m in the Global Volcanism Program database and 3528 m in the KVERT list. In our work, we relied on PeakVisor (https://peakvisor.com), which is one of the most advanced mountain identification and 3D maps tool (also available as a mobile app). Its peaks database has almost a million named summits and is based on DEMs from the European Copernicus program, the United States Geological Survey (USGS), and the Japan Aerospace Exploration Agency (JAXA).
The name, geodetic latitude and longitude, true height, and estimated height of the selected peaks, as well as the date and 420 time of the GOES-17 images used are listed in supplementary Table S1. As previously discussed, the observing conditions show considerable temporal variation, but for a static target one has the luxury of a large number of available images from which to choose one that offers a good view of the peak. For a volcanic plume, one is limited to a few images around the eruption time; however, identifying a high-altitude plume top is also much easier, because reduced atmospheric turbulence and the distinct color of the ejecta usually lead to good contrast against the background. The height error estimates obtained 425 below for volcanic peaks under good 'seeing', nevertheless, represent a lower limit, because the (typically unknown) radial tilt of eruption columns introduces additional uncertainty in the retrievals.
The scatter plot of estimated peak height versus true peak height is given in Fig. 9 (remember, ABI images are upsampled with SPF = 2). For each data point, we performed a sensitivity analysis by shifting the visually determined peak location by ±1 pixel in either direction and calculating heights for the central pixel and its 8-pixel neighborhood. The 430 standard deviation of these nine height values is then used as error bar. The root mean square error (RMSE) computed for the RMSE is 150 m. The error bar on individual retrievals is ~250 m, with a maximum possible height discrepancy of ±400 m due to a 1-pixel uncertainty in peak location; these numbers increase to ~300 m and ±500 m when ABI images are used without up-sampling (SPF = 1). We conclude from these results that in the absence of significant radial tilt, ±500 m is a 435 reasonable uncertainty value for instantaneous height retrievals.

Sheveluch eruption on 8 April 2020
As a final example in Part 1, we demonstrate the side view method on the 8 April 2020 eruption of Sheveluch. A detailed analysis of this case, including a comparison with other height retrieval methods, is given in Part 2. The Sheveluch volcano has three main elements: Old Sheveluch with an elevation of 3307 m, the old caldera, and the active Young Sheveluch with a 440 lava dome at 2589 m surrounded by peaks of about 2800 m elevation. A strong explosive eruption occurred on 8 April 2020 at 19:10 UTC, slightly after sunrise, whose ash plume advected south-southeast from Young Sheveluch. The Kamchatka Volcanic Eruption Response Team issued an orange coded Volcano Observatory Notice for Aviation (VONA 2020-40, http://www.kscnet.ru/ivs/kvert/van/?n=2020-40), reporting a plume height of 9.5-10.0 km as determined by the basic satellite temperature method from Himawari-8 11 µm data. 445 The GOES-17 fixed grid images capturing the eruption are presented in Figure 10. Figure 10a shows the volcano at 19:00 UTC before the eruption, with the volcanic peaks, especially the more northerly Old Sheveluch, clearly recognizable above a low-level stratus layer. Far away peaks of the Sredinny Mountain Range west of Sheveluch can also be seen in the background. By the time of the next full disk image acquired at 19:10 UTC and depicted in Fig. 10b, the eruption column reached its maximum altitude and developed a small umbrella cloud, which was advected slightly to the south-southeast by 450 the prevailing upper-level winds. Notable features in this image are a portion of the long shadow cast by the ash column and the brighter sunlit near side (eastern edge) and the darker shadowed far side (western edge) of the umbrella cloud, brought out by the rising Sun in the east ( # = 86.0°, # = 82.6°).
A simple technique commonly applied to aid change detection in multitemporal imagery is the computation of runningdifference images. The normalized running-difference (RD) image at time t can be defined as 455 where image is a 2D array of reflectances and á ñ indicates the mean of all pixels. The advantage of such a running-difference image, whose mean pixel value is ~0, is that static or quasi-static background features are removed, making identification of dynamic features easier. The RD image calculated from the 19:10 UTC and 19:00 UTC GOES-17 snapshots is plotted in Fig. 10c. Note how the plume and its shadow are accentuated, while Old Sheveluch and the peaks of the Sredinny mountains 460 are removed in this image. Similar or more sophisticated image change detection algorithms (Radke et al., 2005) could later be used for the automated detection of volcanic eruptions in side view imagery, where traditional methods based on brightness temperature differences might be problematic due to the extreme view geometry (e.g. limb darkening or brightening effects).
Finally, Fig. 10d presents the fixed grid view of the plume at 19:10 UTC with the base-relative isoheight lines drawn. 465 The summit of Old Sheveluch is correctly located between the 3 km and 4 km contour lines. Our best estimate plume top position directly above the vent is marked by the blue diamond, which was visually determined considering the expansion and slight advection of the umbrella cloud and which lies halfway between the far-side and near-side edges. The selected top pixel leads to a plume height estimate of ~8 km. Note that the radial expansion of the plume at an approximately constant altitude results in height biases as sketched in Fig. 4. The darker far-side edge of the umbrella, which is located behind the 470 plane of the isoheights, appears at an altitude of ~9 km, while the brighter near-side edge, which is located in front of the isoheights' plane, appears at ~7 km. In contrast, the latitudinal (left-right) expansion of the plume has little effect on the height estimates. The side view plume height of ~8 km is 1.5-2.0 km lower than the height from the temperature method and, based on the discussion above, we think it is closer to the true plume altitude. This discrepancy, caused by well-known retrieval biases near the tropopause temperature inversion, is further investigated in Part 2. 475

Summary
We presented a simple geometric technique that exploits the generally unused near-limb portion of geostationary fixed grid images to estimate the height of a volcanic eruption column. Such oblique angle observations provide an almost orthogonal side view of a vertical column protruding from the Earth ellipsoid and allow height calculation by measuring the angular extent of the column. We demonstrated the technique using data from the ABI instrument, which offers the highest 480 resolution visible imagery and most accurate georegistration among current generation geostationary imagers. The publicly available ABI level 1B data distributed by NOAA also contain all the information required for the calculations.
Thanks to its purely geometric nature, the technique avoids the pitfalls of the traditional brightness temperature method; however, it is mainly applicable to strong eruptions with nearly vertical columns and its coverage is limited to daytime point estimates in the immediate vicinity of the vent. Initial validation of the technique on mountain peaks in Kamchatka and the 485 Kurils indicates that ±500 m is a reasonable preliminary uncertainty value for height estimates of near-vertical eruption columns; this uncertainty compares favorably with the 2-4 km uncertainty typical of state-of-the-art radiometric methods.
The radial expansion of the volcanic umbrella cloud or the radial tilt of a weak eruption column under strong wind shear, however, introduces additional errors that need further characterization. In Part 2 of the paper, we apply the technique to seven recent volcanic eruptions observed by GOES-17 in Kamchatka, the Kuril Islands, and Papua New Guinea, including 490 the 2019 Raikoke eruption, and compare the side view plume height estimates with those from the IR temperature method, stereoscopy as well as ground-based video camera and quadcopter images.

Appendix A A1 Calculation of ABI look vectors and geodetic coordinates
The formulas to compute the geodetic coordinates and the look vector of an ABI fixed grid image pixel are provided here for 495 convenience. The description follows section 5.1.2.8 of the GOES R Series Product Definition and Users' Guide (GOES-R PUG L1B Vol 3 Rev 2.2, 2019), which is the definitive reference. The corresponding formulas for AHI can be found in section 4.4 of the Low Rate Image Transmission (LRIT) / High Rate Image Transmission (HRIT) Global Specification (CGMS, 2013).
The coordinate systems used for image navigation are illustrated in Fig. A1. The Earth Centered Fixed (ECF) coordinate 500 system rotates with the Earth and has its origin at the center of the Earth. Its x axis (ex) passes through the intersection of the Greenwich Meridian and the Equator, while its z axis (ez) passes through the North Pole. Its y axis (ey) is defined as the cross product of the z axis and the x axis to complete the right-handed coordinate system. The satellite coordinate system has its origin at the satellite's center of mass. Its x axis (Sx) points from the satellite to the center of the Earth and its z axis (Sz) is parallel to the ECF z axis (ez). Its y axis (Sy) is aligned with the equatorial axis and 505 completes the right-handed coordinate system.
For point P on the GRS80 ellipsoid, the geocentric latitude is the angle between the local radius vector and the equatorial plane, while the geodetic latitude is the angle between the local surface normal and the equatorial plane. The geodetic latitude, which is the one used in image navigation, is larger than the geocentric latitude at all locations except the poles and the Equator where they are equal (the maximum difference between and is ~0.19º). The geodetic and 510 geocentric longitudes are however the same. Longitude is positive east of and negative west of the Greenwich Meridian.
The notation for the scan angles that describe the position of a pixel within the fixed grid is somewhat confusing though.
In the L1B netCDF data files, the east-west (horizontal) scan variable is called 'x' and the north-south (vertical) scan variable is called 'y', both given in radians. Scan angle 'x' is negative west of and positive east of the subsatellite longitude, and scan angle 'y' is positive north of and negative south of the Equator. The north-south ('y') angle elevates the east-west ('x') scan 515 plane relative to the equatorial plane.
With these caveats in mind, the satellite-to-pixel look vector =`+ ,b , +,c , +,d e is given in the satellite coordinate system as +,b = y cos( ) cos( ) +,c = − y sin( ) 520 . Terrestrial refraction geometry after Noerdlinger (1999). The distant observer (GOES-R) views point ′, which is displaced by distance d and registered in the satellite image at point P. Noerdlinger (1999) provides analytical formulas that relate the known zenith angle of the unrefracted ray to i and ′, from which the horizontal displacement along the surface can be calculated using the local Earth curvature radius } (= 6,371,000 m in the spherical model).