DTM-Based Morphometric Analysis of Scoria Cones of the Chaîne Des Puys (France)—The Classic and a New Approach

remote sensing

Article DTM-Based Morphometric Analysis of Scoria Cones of the Chaîne des Puys (France)—The Classic and a New Approach

Fanni Vörös 1,* , Benjamin van Wyk de Vries 2,Dávid Karátson 3 and Balázs Székely 4

1 Doctoral School of Earth Sciences, ELTE Eötvös Loránd University, Pázmány P. sétány 1/C, H-1117 Budapest, Hungary 2 Laboratoire Magmas et Volcans, Observatoire du Physique du Globe de Clermont, Université Clermont, Auvergne, IRD, UMR6524-CNRS, 63178 Aubiere, France; [email protected] 3 Department of Physical Geography, ELTE Eötvös Loránd University, Pázmány P. sétány 1/C, H-1117 Budapest, Hungary; [email protected] 4 Department of Geophysics and Space Science, ELTE Eötvös Loránd University, Pázmány P. sétány 1/C, H-1117 Budapest, Hungary; [email protected] * Correspondence: [email protected]; Tel.: +36-1-372-2500 (ext. 6701)

Abstract: Scoria cones are favorite targets of morphometric research. However, in-depth, DTM-based studies have appeared only recently, and new methods are being developed. This study provides a classic evaluation of the cones of Chaîne des Puys (Auvergne, France) as well as introduces a more detailed and statistics-based set of properties. Beside the classic parameters, a sectorial approach is applied to the slope distributions calculated from high resolution DTMs for 25 cones of different lithologies, in order to study the various (a)symmetries of the cones. DTM-based morphometric characteristics have been found to be different from classic descriptors, whereas the sectorial approach describes correctly the more and the less regular shapes. The distribution of interquartile ranges of the Citation: Vörös, F.; van Wyk de Vries, sectorial slope distributions is skewed. Sectorization discriminates various types of symmetries: there B.; Karátson, D.; Székely, B. are almost circular cones, but the majority are elongated and have some asymmetry. The relationship DTM-Based Morphometric Analysis of Scoria Cones of the Chaîne des between size parameters reflects the lithology, rather than the age of the cone. The attempt to Puys (France)—The Classic and a relate morphometric parameters to age data is only partially successful: although there is a certain New Approach. Remote Sens. 2021, 13, trend, within the same lithological group, subtle but possibly systematic trends can be detected 1983. https://doi.org/10.3390/ for decreasing morphometric values (e.g., slope) with the age. The regression models indicate rs13101983 various outcomes. Further work is needed to understand all the diverse parameters, especially the lithology–shape relationship, and how symmetry is connected to different factors. Academic Editors: Damien Dhont and Amin Beiranvand Pour Keywords: scoria cone; Auvergne; France; geomorphometry; slope distribution; Mann–Whitney test; Theil–Sen estimator; RANSAC modelling Received: 29 April 2021 Accepted: 18 May 2021 Published: 19 May 2021 1. Introduction Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in Since the beginning of the 2000s, with the appearance of increasingly better resolution published maps and institutional affil- digital terrain models (DTMs), they have not only played a role as a background and illus- iations. tration but have also become increasingly common in various geomorphometric studies. On the one hand, in many cases, the DTM-derived data are more accurate and detailed than the older data obtained from field measurements and based on stereophotogrammetry. In the field, for example, the vegetation can obscure microtopographic elements, which can be seen on the finer DTM. On the other hand, it is perfectly suitable for the study of large Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. groups and geomorphological forms (e.g., [1]). With the help of them, it is not difficult to This article is an open access article analyze even hundreds of forms. distributed under the terms and In this study, we examined one of the simplest volcanic structures, the scoria cone, from conditions of the Creative Commons a geomorphometric point of view. Because they are monogenetic forms, they are perfectly Attribution (CC BY) license (https:// suited to study degradation. Following Porter [2], Wood [3,4], and many other researchers creativecommons.org/licenses/by/ (see previous research), we investigated this degradation in the Chaîne des Puys with 4.0/). different parameters and supplemented them with calculations of scoria cone asymmetry.

Remote Sens. 2021, 13, 1983. https://doi.org/10.3390/rs13101983 https://www.mdpi.com/journal/remotesensing Remote Sens. 2021, 13, x FOR PEER REVIEW 2 of 21

perfectly suited to study degradation. Following Porter [2], Wood [3,4], and many other researchers (see previous research), we investigated this degradation in the Chaîne des Puys with different parameters and supplemented them with calculations of scoria cone asymmetry.

Scoria Cone Morphometry Scoria (or cinder) cones may be the most common volcanic edifices on Earth [3,4]. They are mostly monogenetic, which is a single eruption, or a single eruptive phase is involved in their formation. They can come from a single vent [3,5], or a few focal points

Remote Sens. 2021, 13, 1983 along a fissure [6,7]. The term: “monogenetic” is still widely2 of used 21 [8]. Fornaciai et al. [6] wished to limit the use of this term (based on [9–11]) to cones that developed within a relatively short period of time (hours to months), which are small in volume, and which Scoriahave Cone erupted Morphometry predominantly mafic magmas. Németh (e.g., [12–15]) has shown that mon- Scoria (or cinder) cones may be the most common volcanic edifices on Earth [3,4]. They areogenic mostly volcanoes monogenetic, are which highly is a single variable eruption, orin a both single eruptiveform and phase composition, is involved and that there might inbe their a transition formation. They from can comesimple from scoria a single ventcones [3,5 ],to or long a few- focallived points polygenetic along a volcanoes. Cones are fissure [6,7]. The term: “monogenetic” is still widely used [8]. Fornaciai et al. [6] wished to limitusually the use grouped of this term in (based fields: on [9– they11]) to conescan come that developed into being within a on relatively a flat short surface, in mountainous to- periodpography, of time (hoursor as to parasitic months), which cones are smalldotted in volume, on the and flanks which haveof shield erupted- or stratovolcanoes (e.g., predominantly[16]). mafic magmas. Németh (e.g., [12–15]) has shown that monogenic volcanoes are highly variable in both form and composition, and that there might be a transition from simpleThe scoriapresent cones study to long-lived examines polygenetic the volcanoes. degradati Coneson are of usually scoria grouped cones. Previous studies on the intopic fields: (see they later) can come were into beingbased on on a flat a surface,simple in geometric mountainous model topography, of orfresh as scoria cones [2] and did parasitic cones dotted on the flanks of shield- or stratovolcanoes (e.g., [16]). not Theconsider present studyany examinescone irregularities the degradation e.g of scoria.,. elongated, cones. Previous collapsed studies on, or craterless cones. Porter thewas topic the (see first later) towere establish based on a simple quantitative geometric model relationships of fresh scoria between cones [2] and various measures of cone did not consider any cone irregularities e.g.,. elongated, collapsed, or craterless cones. Portermorphology. was the first He to establish examined quantitative the morphology, relationships between distribution, various measures and ofsize frequency of the cones coneon Mauna morphology. Kea, He examinedHawaii. the In morphology, most papers distribution, (along and with size frequency ours), ofthe the terminology of Porter has cones on Mauna Kea, Hawaii. In most papers (along with ours), the terminology of Porter hasbeen been used used (Figure(Figure1). 1).

Figure 1. 1.Relationships Relationships between between volcanic parameters. volcanic Wparameters.cr = crater width, WcrDcr == crater crater depth,width, Dcr = crater depth, Hco = Hco = cone height, Wco = basal width of cone, α = maximum slope angle (based on [2]). cone height, Wco = basal width of cone, α = maximum slope angle (based on [2]). One of the earliest studies was Colton’s [17] classic account of scoria cone morphology in the San Francisco Volcanic Field, Arizona. He classified them into five degradation stages ofOne cones of and the basaltic earliest flows studies on the basis was of weathering. Colton’s Scott [17] and classic Trask [18 account] studied of scoria cone morphol- theogy morphologic, in the San morphometric, Francisco chemical, Volcanic and Field, radiometric Arizona. data of scoria He classified cones and lava them into five degradation flows in the Lunar Crater volcanic field, Nevada: using maximum cone slope and cone width/conestages of heightcones ratio, and they basaltic derived relativeflows morphologicon the basis ages of for weathering 15 cones. Settle. [Scott19] and Trask [18] studied examinedthe morphologic, parameters of sixmorphometric, volcanic areas (Mauna chemical, Kea (Hawaii), and Mount radiometric Etna (Italy), data Kili- of scoria cones and lava manjaroflows (Tanzania),in the Lunar San Francisco Crater Volcanic volcanic Field (Arizona),field, Nevada: Paricutín (Mexico), using Nunivakmaximum cone slope and cone Island (Alaska)), including average volume, cone height and diameter. One of the most significantwidth/cone works height in scoria ratio cone morphometry, they derived came fromrelative Wood morphologic ([3,4]) who considered ages for 15 cones. Settle [19] conesexamined of the San parameters Francisco Volcanic of six Field, volcanic which had areas been categorized (Mauna by Kea Colton (Hawaii [17] and ), Mount Etna (Italy), Kil- Moore and Wolfe [20] into age groups. Wood examined and compared these and observed trendsimanjaro that were (Tanzania also found), in San five Francisc other scoriao cone Volcanic fields (Lunar Field Crater (Arizona Volcanic), Field Paricutín (Mexico), Nunivak (Nevada),Island ( NewberryAlaska) volcano), including (Oregon), average Wu-ta-lien-chi volume, volcano cone (Manchuria), height Etna and (Italy), diameter. One of the most and Piton de la Fournaise (Réunion)) to determine if the degradation values are common tosignificant other scoria coneworks areas in on scoria Earth. Wood cone noted morphometry that (geological) mapscame did from not provide Wood ([3,4]) who considered adequatecones of information the San for Francisco sufficient resolution Volcanic of certain Field, parameters which (e.g.,had crater been diameter). categorized by Colton [17] and HeMoore claimed and that Wolfe the cone [20] height into (Hco), age height-to-diameter groups. Wood (Wco) examined ratio, and slope and angles compared (α) these and observed trends that were also found in five other scoria cone fields (Lunar Crater Volcanic Field (Nevada), Newberry volcano (Oregon), Wu-ta-lien-chi volcano (Manchuria), Etna (Italy), and Piton de la Fournaise (Réunion)) to determine if the degradation values are common

Remote Sens. 2021, 13, 1983 3 of 21

decrease over time, while the ratio of crater diameter (Wcr) to cone diameter (Wco)(Figure1 ) does not change with decay. Following Wood, a comparative morphometric study along with computer modeling was done by Hooper and Sheridan [21]. Surface erosion processes were modeled on an ideal scoria cone. Maximum slope values were calculated from field surveys, aerial photographs, and distance from contour lines. Digital terrain models (DTM) from the United States Geological Survey were used to supplement these data. Their modeling was based on the parameters described earlier: height, height to diameter ratio (Hco/Wco), crater depth to crater diameter ratio (Hcr/Wcr), and cone slope angle. According to these authors, the basis of relative age determination with comparative morphology is the study of morphological parameters that continuously decrease over time. From the early 2000s, DTMs began to be increasingly used in morphometric studies, as they became more widely available with progressively better resolution. The correlations published in previous studies are mainly idealized for regular, circular, or elliptical cones. In the case of irregular cones, it is of great importance how the diameter is designated: Favalli et al. [22] examined the parameter changes due to lava flow inundation in scoria cone fields, and also the effects of emplacement on a (larger) volcano flank. Similar results were described by Fornaciai et al. [6]. Different resolution DTMs were used in 21 volcanic areas to compare different tectonic settings and climates. They stated that DTM resolution is crucial when selecting scoria cone areas. The 90-m SRTM (Shuttle Radar Topography Mission) is mostly inappropriate, the 30-m ASTER (Advanced Spaceborne Thermal Emission and Reflection Radiometer) can be useful, but only better resolution data are really recommended for scoria cone studies (e.g., 5 or 10 m-resolution LiDAR). Morphometric parameters (V, slope, Hco, Wco, and Wcr) and geometric ratios (Hco/Wco and Wcr/Wco) were calculated for a total of 542 cones.

2. Materials and Methods 2.1. Data Used 2.1.1. Chaîne des Puys Setting The study area here is the Chaîne des Puys, Auvergne, France (Figure2). This is a compact alignment of more than 80 monogenetic volcanoes, long renowned for their diversity and variability [23–25]. The volcanoes are now part of a UNESCO World Heritage site [26]. The cones vary from simple basaltic cones to evolved trachyte domes and have a full range of volcano-geomorphic features such as small to large craters, breached craters, 10s m to 100s m-high cones, and complex flank geometries reflecting short but complex eruptions. We have taken 25 cones (Table1), which are mostly true scoria cones. Some trachyte domes are included (Grand Sarcoui, le Cliersou, Puy de Dôme and Petit puy de Dôme, see in Table1) to explore their similar, but subtly distinct morphometries. The cones and domes share similar construction characteristics, in that they are both formed by accumulation of fragmented material on slopes. The difference is that cone slopes are formed of scoria thrown out by explosions, while dome slopes by the crumbling of lava extruded at the summit. The resulting slopes appear very morphometrically similar because of this parallel evolution [27]. The present geological map of the Chaîne des Puys has been developed progressively since 1975 [28] and provides a classification of the cones into their different compositions and ages that are used in this study [24]. Studies on the Lemptégy volcano and associated cones [29–31] have shown the complex interplay between eruptions and the volcano’s surface morphology. Thus, that area provides a full set of scoria cone types that have also been closely researched for their constructional processes. Remote Sens.Remote 2021, Sens. 13, 2021x FOR, 13 ,PEER 1983 REVIEW 4 of 21 4 of 21

Figure 2. FigureSlope 2.mapSlope of mapthe Chaîne of the Chaîne des Puy dess Puys volcanic volcanic field field and and its its surrounding surrounding (regional(regional 10 10 m m CRAIG CRAIG DTM, DTM, modified modified from [30]) for identificationfrom [30]) for identification of the cones of the(see cones the (seeIDs the in IDsTable in Table 1). 1).

2.1.2.2.1.2. LiDAR LiDAR DTM DTM TheThe selection selection of of DTM DTM (or previously previously DTM) DTM for) morphometric for morphometric studies isstudies always ais key always a key issue. In most of the cases, the experimenter has a limited choice in terms of resolution and issue.accuracy. In most Lower of the resolution cases, DTMs the experimenter underestimate the has slope a limited angles; therefore, choice in it is terms advisable of resolution and toaccuracy. use higher Lower resolution resolution data if available. DTMs underestimate However, previously, the slope a common angles opinion; therefore was , it is ad- visablethat, to in use case higher of scoria resolution cones, being data mostly if available. regular in shape, However, this difference previously is not, crucial,a common as opinion was thethat slopes, in case of the of cones scoria are cones, rather uniform being (thismostly can beregular disproved, in shape, e.g., [32 this]). Unfortunately, difference is not cru- this is not the case, because the irregularities modify the slope distribution introducing a cial, bias.as the Even slopes if the of man-made the cones structures are rather are presentuniform in the(this data, can LiDAR be disproved, DTMs are ae.g., better [32]). Unfor- tunately,approximation this is not of thethe ground case, because surface than the that irregularities of stereophotogrammetry-based modify the slope data.distribution introducing aThis bias. is theEven reason if the we man opted-made to use structures the LiDAR dataare forpresent our studies. in the Thedata, Chaîne LiDAR des DTMs are a betterPuys approximation not only provides anof exemplarythe ground area tosurface study scoria than cone that morphometry of stereophotogrammetry but is covered -based data.extensively by LIDAR (CRAIG: Centre Regional de Informational Geographique—[33]). Two data sets are of particular value, LIDARVEGNE [34], a 0.5 m resolution covering of theTh centralis is the part reason of the chain,we opted and LIDARCLERMONT to use the LiDAR [34 ],data a 5 m-spacingfor our studies. general LIDAR The Chaîne des Puysdata not set only made provides to cover an the exemplary Clermont Auvergne area to Metropolitanstudy scoria area. cone All morphometry of the LIDAR is but is cov- eredopen extensively data, and availableby LIDAR from (CRAIG: CRAIG. These Centre data haveRegional been properly de Informational processed, and, Geographique even – [33]).if Two there aredata man-made sets are structuresof particular (roads, value, some buildings)LIDARVEGNE present [34] in the, a data, 0.5 them resolution overall cover- extent of these non-geomorphic features is negligible in the area of the scoria cones. ing of the central part of the chain, and LIDARCLERMONT [34], a 5 m-spacing general LIDAR data set made to cover the Clermont Auvergne Metropolitan area. All of the LI- DAR is open data, and available from CRAIG. These data have been properly processed, and, even if there are man-made structures (roads, some buildings) present in the data, the overall extent of these non-geomorphic features is negligible in the area of the scoria cones.

2.2. Methods 2.2.1. Classic Morphometry

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Table 1. Summary of the geology, the ages [24] and the calculated classic morphometric parameters of the scoria cones and lava domes of Chaîne des Puys.

Name of Age of Cone Average Slope (◦) Cone ID Lithology [24] H /W W (m) D (m) Cone (ka) [24] co co cr cr According to [35] 2 Petit Sarcoui 31.5 basalt 0.199 299.49 54.38 33.8 Grand 3 12.6 trachyte 0.250 No CR No CR 26.5 Sarcoui Puy des 4 31.5 trachybasalt 0.215 243.73 33.42 31.2 Goules Puy de 6 45 basalt 0.166 310.30 35.13 26.1 Fraisse 7 Puy Pariou 9.5 trachyandesite 0.216 322.42 89.82 31.2 8 le Cliersou 14 basalt 0.225 No CR No CR 24.2 Grande 9 45 basalt 0.211 269.58 40.73 29.4 Suchet 10 Puy de Come 14 trachyandesite 0.237 421.42 75.83 34.7 11 Puy Balmet 45 basalt 0.149 229.16 17.06 23.6 12 Puy Fillu 45 trachybasalt 0.185 215.58 47.52 31.7 13 le Petit Sault 47 basalt 0.191 113.72 19.58 32.2 le Grand 14 47 basalt 0.157 No CR No CR 17.4 Sault 15 Puy Besace 46 basalt 0.186 167.60 32.11 27.5 Puy de 16 43 trachybasalt 0.220 232.45 64.14 33.1 Salomon Puy 17 63.7 trachybasalt 0.177 461.36 51.21 28.5 Montchié Puy de la 18 40 trachybasalt 0.221 295.71 103.89 46.1 Moreno Puy 19 40 trachyandesite 0.214 337.03 39.56 30.8 Laschamp Puy de 20 18.1 trachybasalt 0.226 230.84 38.91 29.0 Mercoeur Puy de 21 20 basalt 0.090 364.45 40.78 25.9 Monteillet Puy de 23 19 trachyandesite 0.217 202.01 48.16 33.5 Montjuger Puy de 24 8.45 trachybasalt 0.223 431.69 152.00 42.2 Lassolas Puy de la 25 9.7 trachyandesite 0.215 303.37 111.56 34.0 Mey Puy de la 26 8.45 trachybasalt 0.298 386.05 152.39 54.3 Vache 27 Puy de Dôme 11 trachyte 0.314 No CR No CR 32.1 Petit puy de 28 11 trachyte 0.259 No CR No CR 27.4 Dôme

2.2. Methods 2.2.1. Classic Morphometry Based on the original methodology described by Settle [19], cone height was defined as the difference between the maximum value of the crater rim, or the summit elevation and the average basal elevation. Cone width or basal diameter (Wco) was calculated as the average of the maximum and the minimum diameters of each cone. These calculation methods were used by other researchers, e.g., [3,4,21,22,35,36]. Crater width or diameter (Wcr) was defined as the average of the maximum and minimum diameters of each crater. Crater depth (Dcr) is the difference between maximum rim or summit elevation and the lowest elevation in the crater. The usefulness of the slope angle (and deriving relative cone ages from it) were identified [18]. Before the appearance of digital terrain models, slope angles were measured in the field, taken from photographs [37] or determined Remote Sens. 2021, 13, 1983 6 of 21

with the help of the contour lines of topographic maps [3,4]. Over time, as the ediﬁce

Remote Sens. 2021, 13, x FOR PEER REVIEWis eroded, it is expected that both the maximum and average slope values6 decrease: of 21 the cone becomes smaller and smoother, as the erodible materials from the higher regions are either transported to the base of the cone (under dry climates) or further away (humid climates).we created Asa maximum a result, radius the maximum of 3000 m. slopeChoosing angle the would (angular) become size of smallerthe sectors than is also that of the initialan important cone. Hasenaka step: if it and is too Carmichael large (e.g. (here, the c afterentral referred angle is to 45°), as H&C) the smaller [35] defined surface how the averagechanges slopewill not angle modify (which the sectorial is another values indicator—in the of same changing way as morphology) if we consider can the be cone calculated, dependingas a whole. onIf it whether is too small a cone (too hasdetailed, or has like no 4°), crater. then Forthe coneslarger correlations which still havecannot a be crater, the averagedetected conebecause slope small angle changes (Save can) is considerably defined by: influence the sector value. As a compromise solution, the sectors were created in every 15° angle, so the entire circle was covered −1 by 24 sectors. The sectors were Snumberedave = tan counterclockwise,[2Hco/(Wco − W sectorcr)], number one is located (1) at 12 o'clock. Thus, the whole circle has been split into twenty-four angular sectors, with where(relative)Hco centralis cone coordinates height, Wco ofis cone(0, 0). width These or abstract diameter, sectors and willWcr henceforthis crater width work or as diameter. For'cookie cones cutter' that shapes have lostin the the next crater steps. to erosion or did not have one originally (Wcr = 0), the averageStep slope #4: the angle sectorial can beshapes calculated created by: in Step #3 are used as follows: (0, 0) centered shapes are shifted to the center (Step #1) of the selected scoria cone. In some cases, we also −1 changed the direction of sector numberSave one= tan (so not[2H atco 12/W o'clock)co] by rotating the pattern. (2) If the crater is not intact, but has collapsed or disappeared due to degradation, or there is 2.2.2.any other Sectorization morphological feature that distorts the circular symmetry, the centerline was rotatedBased to the on thepossible previous axis of literature symmetry. presented The starting above, direction we see was that also generally rotated the if wholethe cone shape of the scoria cone was elongated in any direction. The main direction was chosen has been used for morphometry. This method greatly simplifies the calculations because by the direction of the major axis. At the end of this task, the 24 sectors have a correspond- theing conereal-coordinate can be characterized center, and the by starting only a fewdirection numerical is also values.selected. However, the variability of the coneStep (e.g., #5: the asymmetry) optional presence is not reflected.of a crater Toplays explore a signi this,ficant the role method in the calculations of sectorization: has beenthe slope introduced values are ([ 38greatly] and influenced this paper). by the Sectorization crater, but the is appearance/shape easiest to understand of the cone for circular symmetricis less prominent cones., so The it is sectors advisable are to selected omit thein crater the from radial the direction sectorial usingcalculations. the following In the main stepscase of (Figure an ideal,3): circular cone, a doughnut-shaped mask is applied to the DTM: the middle 1.part covereddetermine by the the center crater ofis theomitted scoria. To cone, do this, the detection of the crater rim has to be 2.done determine at first and the then outline a (concentric) of the scoria circ cone,le is fitted to it. Starting from the center, the farthest point of the crater rim was found radially, and this distance became the radius of the 3. create the initial (basal) sectors, circle to be cut out. After the "cut-out," each sector can be described by four coordinate 4. shift/rotate the starting sectors, pairs of a trapezoid. 5. Stepdetermine #6: the the final crater resulting rim, shapes are obtained as the intersection of the sectors 6.formed create in Step the #5 final and sectors. the shape bounded by the outlines created in Step #2.

FigureFigure 3.3. Example of of the the sectorization sectorization steps steps described described in the in text. the text.The green The green line (Steps line (Steps4–6) indi- 4–6) indicates cates the selected primary direction and the whole setting is rotated to the 12 o’clock position indi- thecated selected by the primarygreen arrow direction (LiDAR and DTM the with whole shaded setting topography). is rotated to the 12 o’clock position indicated by the green arrow (LiDAR DTM with shaded topography).

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Steps #1 and #2 are de facto standard steps, which should be performed not only for sectorization, but also for any morphometric analysis of scoria cones. Nowadays, these steps are (mostly) based on digital terrain models, ideally correlated with geological maps. The base of the cone was defined as the concave break in slope between the cone slope and the surrounding terrain (e.g., as in Figure1). In many cases, this was the low slope plateau, and, in others, a sharp slope change between one cone and the adjacent one (see Figure3, Steps 1 and 2). However, the best and most realistic result can be achieved by supplementing these with field data collection. Therefore, each cone base was manually drawn and checked by several users to verify the results. Step #3: the starting sectors had to be adjusted to scoria cone sizes: we created a starting circle with a radius large enough to fit the largest scoria cone of the area. In this case, we created a maximum radius of 3000 m. Choosing the (angular) size of the sectors is also an important step: if it is too large (e.g., the central angle is 45◦), the smaller surface changes will not modify the sectorial values—in the same way as if we consider the cone as a whole. If it is too small (too detailed, like 4◦), then the larger correlations cannot be detected because small changes can considerably influence the sector value. As a compromise solution, the sectors were created in every 15◦ angle, so the entire circle was covered by 24 sectors. The sectors were numbered counterclockwise, sector number one is located at 12 o’clock. Thus, the whole circle has been split into twenty-four angular sectors, with (relative) central coordinates of (0, 0). These abstract sectors will henceforth work as ‘cookie cutter’ shapes in the next steps. Step #4: the sectorial shapes created in Step #3 are used as follows: (0, 0) centered shapes are shifted to the center (Step #1) of the selected scoria cone. In some cases, we also changed the direction of sector number one (so not at 12 o’clock) by rotating the pattern. If the crater is not intact, but has collapsed or disappeared due to degradation, or there is any other morphological feature that distorts the circular symmetry, the centerline was rotated to the possible axis of symmetry. The starting direction was also rotated if the shape of the scoria cone was elongated in any direction. The main direction was chosen by the direction of the major axis. At the end of this task, the 24 sectors have a corresponding real-coordinate center, and the starting direction is also selected. Step #5: the optional presence of a crater plays a significant role in the calculations: the slope values are greatly influenced by the crater, but the appearance/shape of the cone is less prominent, so it is advisable to omit the crater from the sectorial calculations. In the case of an ideal, circular cone, a doughnut-shaped mask is applied to the DTM: the middle part covered by the crater is omitted. To do this, the detection of the crater rim has to be done at first and then a (concentric) circle is fitted to it. Starting from the center, the farthest point of the crater rim was found radially, and this distance became the radius of the circle to be cut out. After the "cut-out," each sector can be described by four coordinate pairs of a trapezoid. Step #6: the final resulting shapes are obtained as the intersection of the sectors formed in Step #5 and the shape bounded by the outlines created in Step #2.

3. Results 3.1. Results of the Classic Morphometry Parameters Porter [2] selected a group of 30 Mauna Kea cinder cones and reported an empirical relationship of Hco = 0.18 Wco. Bloomﬁeld [39] did the same for 41 cones within a section of the Mexican Volcanic Belt to the east of Paricutín. A mean cone height/width ratio for Holocene cones of 0.21, and a mean Hco/Wco of 0.19 for well-formed and relatively young Pleistocene cones was obtained. Settle [19] used a Hco = 0.2 Wco reference line that characterizes the initial shape of cinder cones formed in both types of volcanic provinces prior to erosive degradation. For our initial work, we made this calculation too. Scoria cone average diameters range from 290 m to ~1500 m with a mean of ~870 m and a median of 830 m. Average cone heights from ~50 m to ~470 m have a mean value of 190 m and a median 180 m (Table1). Cones number 3, 8, 14, 27, and 28 do not have a crater. Remote Sens. 2021, 13, 1983 8 of 21

Remote Sens. 2021, 13, x FOR PEER REVIEW 8 of 21 The distributions of the Wco and Hco values are plotted in Figure4a. The median (yellow) and mean values (blue) do not differ much; however, Wco values slightly, and Hco values are considerably positively skewed (skewness(Wco) = 0.26; skewness(Hco) = 0.99).

Figure 4. (a) Distribution of the basal diameter (Wco) and the height of the cone (Hco) and (b) distribution of the Hco/Wco Figure 4. (a)values Distribution of the studied of 25the volcanic basal cones diameter of the Chaîne (Wco) des and Puys. the height of the cone (Hco) and (b) distribution of the Hco/Wco values of the studied 25 volcanic cones of the Chaîne des Puys. Similar to other authors (e.g., [2,4,19,21]), we also considered the relationship of the basal diameter and the height of the cone via Hco/Wco values, and we also analyzed this Similarrelationship. to Theother distribution authors of (e.g. the Hco, /[2,4,19,21]Wco values is), plotted we also in Figure considered4b. The distribution the relationship of the − basal ofdiameter these ratio and values the does height not show of the skewness cone (= via0.146) Hco/ andWco the values, mean (=0.21)and we and also the analyzed this median (=0.215) values are very close to each other. It also shows a great similarity to the relationshivalues usedp. The by aforementioneddistribution authorsof the H (0.18,co/W 0.19,co values 0.21). is plotted in Figure 4b. The distribution of these ratioWe emphasize values thatdoes each not of show the calculations skewness listed (= and –0.146) deﬁned and above the describes mean a(= 0.21) and the mediansymmetric (= 0.215) cone values (with or are without very a crater).close to If weeach consider other these. It also single-value shows descriptors a great similarity to the of the geometry, the distributions of the related values of Chaîne des Puys follow this values used by aforementioned authors (0.18, 0.19, 0.21). trend. Cone height (Hco) versus cone basal diameter (Wco) is plotted in Figure5a (number Weof cones emphasize = 25): the dashedthat each line representsof the calculations Settle’s Hco = listed 0.2 Wco and, the soliddefined line represents above describes a sym- metricour cone data’s (with calculated or without trendline Haco crater).= 0.2244 IfW cowe. In consider Figure5b, the theseWcr/ singleWco ratio-value for n = descriptors 20 of the cones can be seen. A reference (dashed line) is that of Porter’s Wcr = 0.40 Wco and is plotted geometry,for comparison. the distributions of the related values of Chaîne des Puys follow this trend. Cone heightAccording (Hco) versus to Dohrenwend cone basal et al. [diameter40] and Hooper (Wco and) is Sheridanplotted [21in], Figure the measured 5a (number of cones ± ± = 25):values the dashed of the various line represents morphometric Settle’s parameters Hco have = 0.2 an W uncertaintyco, the solid of 10%line to represents15%. our data’s Our results also show similar scatter. calculated trendline Hco = 0.2244 Wco. In Figure 5b, the Wcr/Wco ratio for n = 20 cones can be In Figure5c,d, the Hco/Wco ratio and the average slope (Equations (1) and (2)) are seen. plottedA reference according (dashed to scoria line) cone is age that colored of Porter’s by lithology. Wcr In= both0.40 cases,Wco and the trachyticis plotted for comparison. cones/domes (red points) are situated in a compact group: these are some of the youngest landforms and have high Hco/Wco ratio. Basaltic cones are characterized by lower values; Accordingmost of them areto belowDohrenwend 0.2 and 30◦—separated et al. [40] from and the Hooper trachyandesitic and conesSheridan and roughly [21], the measured valuesseparated of the various from the morphometric trachybasalts. In aparameters few cases, the have calculation an uncertainty method deﬁned of ±10% by to ±15%. Our resultsHasenaka also show and Carmichael similar scatter. [35] results in unexpectedly high average slope angles (even above 50◦). These values far exceed the values published in previous research (e.g., [21]). These values likely resulted because, in the case of cones where the crater rim is not intact, one of the two crater diameter values (minimum/maximum) is missing, so no average value could be calculated.

Figure 5. (a) Hco/Wco ratio, (b) Wcr/Wco ratio, (c) Hco/Wco ratio according to ages (black: basalt, green: trachybasalt, yellow: trachyandesite, red: trachyte), (d) average slope values according to ages (black: basalt, green: trachybasalt, yellow: trachyandesite, red: trachyte).

Remote Sens. 2021, 13, x FOR PEER REVIEW 8 of 21

Figure 4. (a) Distribution of the basal diameter (Wco) and the height of the cone (Hco) and (b) distribution of the Hco/Wco values of the studied 25 volcanic cones of the Chaîne des Puys.

Similar to other authors (e.g., [2,4,19,21]), we also considered the relationship of the basal diameter and the height of the cone via Hco/Wco values, and we also analyzed this relationship. The distribution of the Hco/Wco values is plotted in Figure 4b. The distribution of these ratio values does not show skewness (= –0.146) and the mean (= 0.21) and the median (= 0.215) values are very close to each other. It also shows a great similarity to the values used by aforementioned authors (0.18, 0.19, 0.21). We emphasize that each of the calculations listed and defined above describes a symmetric cone (with or without a crater). If we consider these single-value descriptors of the geometry, the distributions of the related values of Chaîne des Puys follow this trend. Cone height (Hco) versus cone basal diameter (Wco) is plotted in Figure 5a (number of cones = 25): the dashed line represents Settle’s Hco = 0.2 Wco, the solid line represents our data’s calculated trendline Hco = 0.2244 Wco. In Figure 5b, the Wcr/Wco ratio for n = 20 cones can be seen. A reference (dashed line) is that of Porter’s Wcr = 0.40 Wco and is plotted for comparison. According to Dohrenwend et al. [40] and Hooper and Sheridan [21], the measured Remote Sens. 2021, 13, 1983 9 of 21 values of the various morphometric parameters have an uncertainty of ±10% to ±15%. Our results also show similar scatter.

co co cr co co co FigureFigure 5. (5.a) (Haco) /HWco/Wratio, ratio, (b) W(bcr)/ WWco/Wratio, ratio, (c) H co(c/)W Hco /ratioW ratio according according to ages to (black: ages basalt, (black: green: basalt trachybasalt,, green: yellow:trachybasalt, trachyandesite, yellow: red: trachyandesite, trachyte), (d) average red: slope trachyte values), according (d) average to ages slope (black: values basalt, according green: trachybasalt, to ages yellow: trachyandesite,(black: basalt, red: green: trachyte). trachybasalt, yellow: trachyandesite, red: trachyte).

As was emphasized earlier, most previous authors considered single descriptors because they did not have access to adequate input data. The advantage of a digital terrain model-based morphometric research is that it is not necessary to use approximate or single values, but the terrain model can be used to easily obtain values closer to reality (depending on the resolution). Utilizing this advantage, we examined the average slope values of the scoria cones. In addition (following H&C [35]), we separated cones with craters as well. The sectorization method that we introduced requires the removal of the craters anyway (see Section 2.2.2), so we examined the mean slope angles in both cases—that is, either when calculating the entire cone (including the crater) or when we excluded the crater. In the case of the whole cone, the average slope angle is 19.62◦ ± 4.18◦, while, without a crater, 19.57◦ ± 3.96◦. The average difference between the two groups is 0.04◦ ± 0.8◦. In Figure6, two average slope values are plotted. Red columns represent the mean slope calculated using the H&C [35] equations and green columns represent mean slope derived from the DTM (whole cone values). The order of the scoria cones is as follows: we examined the results obtained with the two types of calculations and arranged them in ascending order according to their differences. Cones framed in black are those that do not have a crater (four of them are domes): we can see that the difference between them is the smallest. Mean slope values calculated using the equations of H&C [35] are higher than values derived from DTM: in some cases, these are unrealistically high values. This overestimation has important implications: if we use the more realistic average slopes, the age/slope relationship seems to be more reliable (see 4. Discussion). Remote Sens. 2021, 13, x FOR PEER REVIEW 9 of 21

In Figure 5c,d, the Hco/Wco ratio and the average slope (Equations (1) and (2)) are plotted according to scoria cone age colored by lithology. In both cases, the trachytic cones/domes (red points) are situated in a compact group: these are some of the youngest landforms and have high Hco/Wco ratio. Basaltic cones are characterized by lower values; most of them are below 0.2 and 30°—separated from the trachyandesitic cones and roughly separated from the trachybasalts. In a few cases, the calculation method defined by Hasenaka and Carmichael [35] results in unexpectedly high average slope angles (even above 50°). These values far exceed the values published in previous research (e.g., [21]). These values likely resulted because, in the case of cones where the crater rim is not intact, one of the two crater diameter values (minimum/maximum) is missing, so no average value could be calculated. As was emphasized earlier, most previous authors considered single descriptors because they did not have access to adequate input data. The advantage of a digital terrain model-based morphometric research is that it is not necessary to use approximate or single values, but the terrain model can be used to easily obtain values closer to reality (depending on the resolution). Utilizing this advantage, we examined the average slope values of the scoria cones. In addition (following H&C [35]), we separated cones with craters as well. The sectorization method that we introduced requires the removal of the craters anyway (see 2.2.2.), so we examined the mean slope angles in both cases—that is, either when calculating the entire cone (including the crater) or when we excluded the crater. In the case of the whole cone, the average slope angle is 19.62° ± 4.18°, while, without a crater, 19.57° ± 3.96°. The average difference between the two groups is 0.04° ± 0.8°. In Figure 6, two average slope values are plotted. Red columns represent the mean slope calculated using the H&C [35] equations and green columns represent mean slope derived from the DTM (whole cone values). The order of the scoria cones is as follows: we examined the results obtained with the two types of calculations and arranged them in ascending order according to their differences. Cones framed in black are those that do not have a crater (four of them are domes): we can see that the difference between them is the smallest. Mean slope values calculated using the equations of H&C [35] are higher than values derived from DTM: in some cases, these are unrealistically high values. This Remote Sens. 2021, 13, 1983 overestimation has important implications: if we use the more realistic average10 of 21 slopes, the age/slope relationship seems to be more reliable (see 4. Discussion).

Figure 6. DFigureifference 6. Difference in average in average slope slope calculations: calculations: according according to H&CH&C [35 [35]] (red) (red) and and based based on DTM on (green). DTM (green). The former The former (parameter-based) method obviously overestimates the average slopes, sometimes extremely. Remote(parameter Sens. 2021, -13based), x FOR method PEER REVIEW obviously overestimates the average slopes, sometimes extremely. 10 of 21

Following Hooper and Sheridan [21], Mann–Whitney tests were performed to compare Followingthe parameter Hooper distributions and Sheridan between four [21] groups, Mann with–Whitney different tests lithologies were (Figure performed5c,d, to compare alsothe seeparameter Table1). Using distributions a signiﬁcance between level of αfour= 0.05, groups the statistical with testsdifferent were done litholog for theies (Figure Hco/Wco, the basaltic, trachybasaltic, trachyandesitic, and trachytic group display progres- 5c,d, fouralso groups see Table comparing 1). Using age, H aco significance/Wco and mean level slope of angle α = (DTM) 0.05, the (Figure statistical7). For H testsco/Wco were, done sivelythe higher basaltic, values, trachybasaltic, to some extent trachyandesitic, correlating and with trachytic progressively group display younger progressively ages, whereas, for the four groups comparing age, Hco/Wco and mean slope angle (DTM) (Figure 7). For for meanhigher slope, values, this to some relationship extent correlating is less withpronounced, progressively and younger only ages,the basaltic whereas, cones for (being less steep)mean slope, can thisbe distinguished relationship is less from pronounced, the latter and three only the lithological basaltic cones groups (being, lesswhich show steep) can be distinguished from the latter three lithological groups, which show higher higher but overlapping slope values. The Hco/Wco, values also significantly separate the but overlapping slope values. The Hco/Wco, values also signiﬁcantly separate the trachyte trachytefrom from the trachyandesite the trachyandesite lithologies, lithologies, and, according and to, the according average slope, to the the average basaltic group slope, the ba- salticis gro separatedup is separated from the trachybasaltic from the trachybasaltic one. one.

Figure 7. DistributionFigure 7. Distribution of lithology of lithology groups groups according according to (a to) ages, (a) ages, (b) ( bH) coH/coW/coW, coand, and (c) ( cmean) mean slope slope (black: (black: basalt,basalt, green: green: trachybasalt, yellow:trachybasalt, trachyandesite, yellow: trachyandesite, red: trachyte) red:. trachyte).

3.2. Results3.2. Results of Sectorization of Sectorization Calculations Calculations For the sectorization division, minimum, maximum, mean, median, and mode values For the sectorization division, minimum, maximum, mean, median, and mode values were calculated for each sector (24) of the 25 examined cones. There are two cones that were havecalculated been so destroyedfor each (eithersector by (24) volcanogenic of the 25 processes examined or post-eruptional cones. There erosion) are two that cones that have certainbeen so sectors destroyed are missing (either (cones by 2 volcanogenic and 26). As was processes explained in or Section post -2.2.2eruptional, the starting erosion) that certain(main) sectors direction are missing of the sectors (cones was inﬂuenced2 and 26). by As the was shape explained of the cone. in The 2.2.2., other the dominant starting (main) directiondirection of the is perpendicularsectors was toinfluenced this main direction.by the shape These of two the imaginary cone. The lines other gave dominant the di- smallest and largest diameters of the scoria cone in most cases (used in Results), and they rectionsplit is theperpendicular whole cone into to quarters. this main With direction. these quarters, These the two cone i canmaginary be well characterizedlines gave the smallest and largest diameters of the scoria cone in most cases (used in Results), and they split the whole cone into quarters. With these quarters, the cone can be well characterized in terms of symmetry (see later). In Figure 8, the histograms of the median slope angles are plotted. Theoretically, a perfectly regular cone would have a single slope value for all sectors; in other words, the histogram would show a single, sharp peak. Consequently, the fewer bins that are filled with values, the more regular, rotationally symmetric is the cone. For instance, Cone 10 has only median slope values between 25° and 30° in the sectors. This cone is really close to regular, even if it is somewhat elongated.

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in terms of symmetry (see later). In Figure8, the histograms of the median slope angles are plotted.

Figure 8. The histograms of the median slope angles of the sectors for the 25 cones studied of the Chaîne des Puys. Figure 8. The histograms of the median slope angles of the sectors for the 25 cones studied of the Chaîne des Puys. Theoretically, a perfectly regular cone would have a single slope value for all sectors; It inis otherquite words, straightforward the histogram would to study show the a single, interquartile sharp peak. Consequently,values (differences the fewer of the values bins that are ﬁlled with values, the more regular, rotationally symmetric is the cone. For at 75% instance,and 25% Cone of 10the has distribution) only median slope to assess values betweenhow compact 25◦ and 30 the◦ in distributions the sectors. This are. As many authorscone assumed is really closethat to the regular, slope even distribution if it is somewhat is elongated.typically rather peaked because of the apparent symmetryIt is quite of straightforward the cones, towe study found the interquartile it informative values to (differences present of the the interquartile values distri- at 75% and 25% of the distribution) to assess how compact the distributions are. As butionsmany of the authors slopes assumed (whole that cone) the slope for distribution the studied is typically individual rather edifices. peaked because In the of case of highly regularthe cones apparent, the symmetry median of values the cones, of we interquartile found it informative range to should present the be interquartile relatively low (all sectors shoulddistributions be similar) of the slopes, and (whole the standard cone) for the deviation studied individual of the ediﬁces.interquartile In the case ranges of should be highly regular cones, the median values of interquartile range should be relatively low (all close tosectors zero should(all distributions be similar), and of the the standard sectors deviation should of the be interquartile similar). rangesFigure should 9 shows be the results for the closestudied to zero cones. (all distributions of the sectors should be similar). Figure9 shows the results for the studied cones.

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Remote Sens. 2021, 13, 1983 12 of 21

Figure 9. Left: the median values of the interquartile ranges of the sectorial slope distributions for the 25 cones studied of the Chaîne des Puys; right: the standard deviations of the same.

However, it is important to clarify that the slope angle distribution is a statistical result, independentFigure 9. Left: the from median the values spatial of the (directional) interquartile ranges contexts of the; sectorial thus, it slope does distributions not give for an accurate Figurepicture 9.the Left: of 25 the conesthe median overall studied values ofsymmetry the of Chaîne the interquartile des of Puys;the cones. ranges right: the of To the standard quantify sectorial deviations slope the distributions symmetry/asymmetry of the same. for of the the 25 cones studied of the Chaîne des Puys; right: the standard deviations of the same. cones, theHowever, slopes of it sectorized is important parts to clarify have that been the slopefurther angle evaluated. distribution Mo isst a of statistical the findings are discussedHowever,result, in independent it the is important next fromsection to theclarify spatialin thethat (directional)context the slope of angle contexts;the distributionlithology thus, it doesandis a statistic notage give ofal anthe re- accurate cone. sult, independentpicture of thefrom overall the spatial symmetry (directional) of the cones. contexts To quantify; thus, it thedoes symmetry/asymmetry not give an accurate of the picture4. Discussion cones,of the overall the slopes symmetry of sectorized of the partscones. have To quantify been further the symmetry/asymmetry evaluated. Most of the of ﬁndings the are cones, thediscussed slopes of in sectorized the next section parts have in the been context further of the evaluated. lithology Mo andst ageof the of findings the cone. are discussedIn inFigure the next 10a section,b, boxplot in the context diagrams of the oflithology the median and age slopeof the cone.values of the sectors can be seen, 4.supplemented Discussion with information on lithology and age. In Figure 10a, the cones were 4.plotted Discussion accordingIn Figure 10 toa,b, increasing boxplot diagrams median of the values. median The slope figure values ofsuggests the sectors the can same be seen, conclusion supplemented with information on lithology and age. In Figure 10a, the cones were plotted thatIn we Figure can 10adraw,b, boxplot from Fig diagramsure 7c: of the the slope median values slope arevalues constrained of the sectors by can the be lithological char- according to increasing median values. The ﬁgure suggests the same conclusion that we seen, supplemented with information on lithology and age. In Figure 10a, the cones were acteristicscan draw rather from Figurethan 7byc: theage. slope In valuesorder are to constrained emphasize by thethis lithological conclusion characteristics in Figure 10b, the plotted according to increasing median values. The figure suggests the same conclusion conesrather were than grouped by age. by In the order lithology to emphasize and this then, conclusion within inthe Figure groups, 10b, thearranged cones were according to that we can draw from Figure 7c: the slope values are constrained by the lithological char- grouped by the lithology and then, within the groups, arranged according to their age. acteristicstheir age. rather than by age. In order to emphasize this conclusion in Figure 10b, the cones were grouped by the lithology and then, within the groups, arranged according to their age.

Figure 10. Boxplot diagrams of the median slope values of the sectors. Cones plotted according to (a) increasing median Figure 10.Figure Boxplot 10. Boxplotdiagrams diagrams of the of median the median slope slope values values of of the sectors.sectors. Cones Cones plotted plotted according according to (a) increasing to (a) increasing median median values andvalues (b) andaccording (b) according to lithology to lithology and then and ages then ages(black: (black: basalt, basalt, green: green: trachybasalt, trachybasalt, yellow: yellow: trachyandesite, trachyandesite, red: red: tra- trachyte). valueschyte) and. (b) according to lithology and then ages (black: basalt, green: trachybasalt, yellow: trachyandesite, red: trachyte).

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Remote Sens. 2021, 13, 1983 13 of 21

Figure 10b suggests that, for basaltic and trachytic cones, there no relationship with the age. In the case of trachyte, the ages are so close to each other that we cannot expect Figure 10b suggests that, for basaltic and trachytic cones, there no relationship with any correlation. In the case of basalt, there are two age groups: an older one of 45–47 ka, the age. In the case of trachyte, the ages are so close to each other that we cannot expect any and a younger but more distributed group ranging from 20 ka to 31.5 ka. Concerning correlation. In the case of basalt, there are two age groups: an older one of 45–47 ka, and a trachybasalt and trachyandesite cones, especially the latter ones, a decreasing trend can younger but more distributed group ranging from 20 ka to 31.5 ka. Concerning trachybasalt andbe observed. trachyandesite For trachybasalts cones, especially, the regression the latter ones,line H aco decreasing/Wco = –0.0014 trend × age can + be0.2669 observed. (R² = 0.5912) and Save = –0.117 × age + 24.404 (R² = 0.4537) can be obtained. For trachyandesitic2 For trachybasalts, the regression line Hco/Wco = −0.0014 × age + 0.2669 (R = 0.5912) and cones, the correlation is even weaker.2 Save = −0.117 × age + 24.404 (R = 0.4537) can be obtained. For trachyandesitic cones, the correlationThe distribution is even weaker. of points in Figure 5a,b indicates some heteroscedasticity, i.e., some pointsThe seem distribution to follow of a pointsdifferent in Figuretrend.5 a,bIn order indicates to study some this, heteroscedasticity, the Theil–Sen i.e.,estimator some points[41,42]seem and RANSAC to follow a algorithm different trend. [43] ha Inve order been to applied study this, to the the data. Theil–Sen estimator [41,42] and RANSACFor the W algorithmco vs. Hco plot [43], havethe results been appliedare very to similar the data. to the linear regression (Figure 11). For the Wco vs. Hco plot, the results are very similar to the linear regression (Figure 11).

Figure 11. ModeliModelingng of Wco––HHcoco relationshiprelationship using using the the ordinary ordinary least least squares squares method method (OLS), (OLS), Theil Theil–– Sen estimation and the RANSAC algorithm. The values indicated are the ages of thethe cones,cones, andand the IDsthe IDs (see (see Table Table1) are 1) indicated are indicated in brackets. in brackets. The colorThe c codeolor code is the is same the same as for as Figures for Figure5,7 ands 5, 710 and. 10.

The three models (ordinary least squares (OLS), Theil–SenTheil–Sen estimator,estimator, andand RANSACRANSAC algorithm) do not differ too much much.. H However,owever, it can be observed that robust robust models (Theil (Theil–– Sen and RANSAC) gave more similar results. It is interesting to notenote that allall trachytictrachytic cones/domes and and two two basaltic basaltic ones ones appear appear as as outliers outliers in in the the RANSAC RANSAC model. model. Although it is expected to describedescribe thethe samesame phenomenon,phenomenon, analyzinganalyzing thethe averageaverage slopes in in the the light light of of the the HHco/coW/coW ratioco ratio,, the theheteroscedastic heteroscedastic character character is more is more obvious obvious (Fig- (Figureure 12). 12This). This figure ﬁgure dem demonstratesonstrates a few a fewinteresting interesting behaviors: behaviors: 1. In this case, the least squares and the Theil–Sen regression give similar results, whereas the RANSAC solution enhances the special behavior of some points too. 2. The trachytic domes are well off any regression line; consequently, RANSAC identiﬁes them as outliers in this relationship as well. 3. The inliers of the RANSAC solution encompass six basaltic cones, a few trachybasaltic, and two trachyandesitic cones. 4. Even if some of them are found to be outliers by RANSAC, trachybasaltic cones are mostly aligned with the trend. However, trachyandesitic cones form rather a compact group if they are considered separately. 5. It is interesting to note that six basaltic cones (out of 8) are strongly aligned, indicating a stronger relationship. 6. The behavior of the basaltic cones motivated us to analyze them separately as well, in particular since this is the largest lithological group of eight cones. The result is presented in Figure 13a.

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Remote Sens. 2021, 13, 1983 14 of 21 Remote Sens. 2021, 13, x FOR PEER REVIEW 14 of 21 Figure 12. Modeling of average slope Wco/Hco relationship using the ordinary least squares method (OLS), Theil–Sen estimation, and RANSAC algorithm. The values indicated are the IDs of the cones.

1. In this case, the least squares and the Theil–Sen regression give similar results, whereas the RANSAC solution enhances the special behavior of some points too. 2. The trachytic domes are well off any regression line; consequently, RANSAC identi- fies them as outliers in this relationship as well. 3. The inliers of the RANSAC solution encompass six basaltic cones, a few trachybasaltic, and two trachyandesitic cones. 4. Even if some of them are found to be outliers by RANSAC, trachybasaltic cones are mostly aligned with the trend. However, trachyandesitic cones form rather a compact group if they are considered separately. 5. It is interesting to note that six basaltic cones (out of 8) are strongly aligned, indicating a stronger relationship. 6. The behavior of the basaltic cones motivated us to analyze them separately as well,

Figurein particular 12. Modeling since of average this is slopeslope the WWco colargest//HHcoco relationshiprelationship lithological usingusing thegroupthe ordinary ordinary of least eightleast squares squares cones. method The result is (OLS),presentedmethod Theil–Sen (OLS), in Theil estimation,Fig–ureSen estimation 13a and. RANSAC, and RANSAC algorithm. algorithm. The values The indicated values indicated are the IDs are of the the IDs cones. of the cones.

Figure 13.Figure Modeling 13. Modeling of average of average slope slope and and HcoH/Wco/coW relationshipco relationship of ((aa)) basalt basalt (b )( trachybasaltb) trachybasalt cones usingcones the using ordinary the leastordinary least squares methodsquares method(OLS), (OLS),Theil– Theil–SenSen estimation estimation,, and and RANSAC RANSAC algorithm.algorithm. The The values values indicated indicated are the are ages the of theages cones, of the and cones, and the IDs (seethe Table IDs (see 1) Table are 1indicated) are indicated in brackets. in brackets. The six cones are aligned and dominate the whole group as expected: RANSAC identiﬁesThe six cones them asare inliers, aligned whereas and dominate two other basalticthe whole cones group are categorized as expected: as outliers. RANSAC iden- tifiesFigure them 13asa inliers, also indicates whereas the agestwoof other the cones. basaltic The cones age distribution are categorized does not as follow outliers. a Figure regular trend. We also tested the same selection for trachybasaltic cones (Figure 13b); in this case, the relationship is also questionable because of the appearance of several cones in the center. To examine the symmetry of the cones, the cones were analyzed along two symmetry axes (see the method above), and the different slope values obtained within the quarters were examined. The cones divided into 24 sectors were examined as follows: we ﬁrst

Figure 13. Modeling of averageexamined slope and the H differenceco/Wco relationship between of the(a) basalt median (b) slope trachybasalt values cones in sectors using 1–12the ordinary and 13–24 least (“left- squares method (OLS), Theilto-right”)–Sen estimation with, theand helpRANSAC of Mann–Whitney algorithm. The values tests indicated (signiﬁcance are the level agesα of= the 0.05). cones, Then, and we the IDs (see Table 1) are indicatedexamined in brackets. the same values in a different grouping: 7-8-9-..16-17-18 and 19-20-21-..-4-5-6 (“up-down”) as it is shown in the right above of Figure 14b. The six cones are aligned and dominate the whole group as expected: RANSAC iden- tifies them as inliers, whereas two other basaltic cones are categorized as outliers. Figure

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13a also indicates the ages of the cones. The age distribution does not follow a regular trend. We also tested the same selection for trachybasaltic cones (Figure 13b); in this case, the relationship is also questionable because of the appearance of several cones in the center. To examine the symmetry of the cones, the cones were analyzed along two symmetry axes (see the method above), and the different slope values obtained within the quarters were examined. The cones divided into 24 sectors were examined as follows: we first examined the difference between the median slope values in sectors 1–12 and 13–24 ("left- to-right") with the help of Mann–Whitney tests (significance level α = 0.05). Then, we examined the same values in a different grouping: 7-8-9-..16-17-18 and 19-20-21-..-4-5-6 ("up- down") as it is shown in the right above of Figure 14b. We examined whether the "left-to-right" and "up-down" directions relative to the Remote Sens. 2021, 13, 1983 15 of 21 starting (main) direction are significantly different. Two p-values, pLR and pUD were obtained for each cone, which are plotted in Figure 14a.

Figure 14. Figure(a) probability 14. (a) probability plot of plot Mann of Mann-Whitney-Whitney tests tests comparing comparing thethe distributions distributions of the of left-rightthe left- andright up-down and up (forward-down (forward backward)backward) sectorial sectorialparts of parts each of considered each considered cones. cones. The The higher higher the value,value the, the more more symmetric symmetric the cone the is cone in that is directional in that directional sense; (b) sense;cone 14 (b), conewhic 14,h is which the ismost the mostsymmetric symmetric one one according according toto thethe chart chart in thein the left panel.left panel. The deﬁnition The definition of the two of the two symmetry symmetryaxes is right axes above. is right above. We examined whether the “left-to-right” and “up-down” directions relative to the Thestarting closer (main) a cone direction is to are one signiﬁcantly of the axes, different. the more Two asymmetricalp-values, pLR and it pisUD inwere that respect. Conesobtained with one for eachp-value cone, low which and are the plotted other in Figure high 14(closea. to 1) elongated in some (denoting the high valueThe closer axis) a conedirection is to one and of are the axes,elliptical. the more An asymmetricalideal, fully symmetrical it is in that respect. cone of rotation wouldCones withbe in one thep-value (1.0, low1.0) andposition. the other In high Figure (close 14b to 1), Cone elongated 14 is in shown some (denoting (LiDAR the DTM with high value axis) direction and are elliptical. An ideal, fully symmetrical cone of rotation shadedwould topography). be in the (1.0, Both 1.0) p position.-values Inof Figurethis cone 14b, are Cone high 14 is (0.93 shown and (LiDAR 0.67), DTM which with means that they areshaded symmetric topography)., not Bothonly pon-values the "left of this-to cone-right are direction" high (0.93 andand 0.67), "up- whichdown meansdirection" separately,that but they also are symmetric,close to a notregular only oncone. the "left-to-right direction" and "up-down direction" Theseparately, important but also message close to aof regular Figure cone. 14b is that, in most of the cases, the symmetry of The important message of Figure 14b is that, in most of the cases, the symmetry of the slopethe slope distributions distributions cannot cannot be assumedassumed along along the hypothesisedthe hypothesised symmetry symmetry axes. This axes. This observationobservation motivated motivated us us to to visualize visualize the twotwo most most important important statistical statistical values values charac- character- izing terizingthe sectors the sectors in the in light the light of the of the classic classic parameters parameters.. In In order order to beto ablebe able to compare to compare the differentlythe differently calculated calculated slope slope angle angle values values clearly and and simply, simply, we plotted we plotted them on them radar on radar chartscharts (Figure (Figure 15). 15In). the In theoutermost outermost circle, circle, the the sectors sectors fromfrom 1 1 to to 24 24 are are indicated indicated in a in a coun- counterclockwise direction. The inner fainter circles show the slope values in 5-degree terclockwisesteps. Four direction. types of values The inner are displayed: fainter twocircles are boundshow tothe sectors slope (reddish), values twoin 5 to-degree the steps. Four entiretypes cone of values (blueish). are The displayed: sectorial data two are theare mean bound and to median sectors values (reddish), of the slopes two in to the the entire cone respective(blueish). direction The sectorial (within onedata sector). are the Since mean all sectors and have median the same values value, of the the regular slopes in the respectivecircles direction are indicating (within calculated one sector). mean slope Since (for all the sectors whole cone) have and the the same result value, of mean the regular slope calculated according to the method of Hasenaka and Carmichael [35]. In most cases, circlesthe are latter indicating shows a much calculated larger value mean than slope those (for obtained the whole by the othercone) three and calculations, the result of mean except for cones without craters (see also above).

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slope calculated according to the method of Hasenaka and Carmichael [35]. In most cases, the latter shows a much larger value than those obtained by the other three calculations, except for cones without craters (see also above). It is clear that, in most of the cases, the H&C [35] method overestimates all types of calculated slopes. On the other hand, naturally, the mean and median slopes are strongly related to the sectorial distribution. For the roughly regular cones, they are close to each other (e.g., Cone 10 and Cone 23). However, in the other way round, if we consider the Remote Sens. 2021, 13, 1983 16 of 21 symmetry of the curves, we will not necessarily find a circularly symmetric cone (e.g., Cone 12 or Cone 13).

FigureFigure 15. Sectorial 15. Sectorial values of values mean slopeof mean angles slope (light angles red), of(light median red), of of the median slope angles of the (red). slope Blue angles circles (red). are calculated valuesBl forue the circle entires are cone: calculated light blue values is the DTM-based for the entire mean cone: slope, light whereas blue theis the blue DTM (outer)-based one ismean calculated slope, according to the methodwhereas of Hasenaka the blue and (outer) Carmichael one is calculated [35]. Note the according great variety to the of themethod sectorial of Hasenaka values. and Carmichael [35]. Note the great variety of the sectorial values. It is clear that, in most of the cases, the H&C [35] method overestimates all types of Last but not calculatedleast, we consider slopes. On the the age other–geometry hand, naturally, relationship the mean as well. and medianFigure slopes16 sum- are strongly marizes the classicrelated slope to value the sectorial calculat distribution.ion by Hasenaka For the roughlyand Carmichael regular cones, [35] they, the areslope close to each distribution calculationsother (e.g., of this Cone paper 10 and, and Cone the 23). ages However, of the cones. in the Similarly other way to round, the previous if we consider the symmetry of the curves, we will not necessarily ﬁnd a circularly symmetric cone (e.g., Cone 12 or Cone 13). Last but not least, we consider the age–geometry relationship as well. Figure 16 summarizes the classic slope value calculation by Hasenaka and Carmichael [35], the slope distribution calculations of this paper, and the ages of the cones. Similarly to the previous Remote Sens. 2021, 13, x FOR PEER REVIEW 17 of 21 Remote Sens. 2021, 13, 1983 17 of 21

observations,observations, the the figure ﬁgure suggests suggests thatthat DTM-based DTM-based slope slope values values give a give more a reliable more picture reliable picture aboutabout the cones the cones,, assuming assuming a a certain certain level level of of regularity. regularity.

Figure 16. FigureSummar 16. ySummary of slope of values slope valuesbased basedon H&C on H&C equa equationstions [35] [35 (dots)] (dots) and and DTM DTMss (triangles)(triangles) according according to its to ages. its ages. Col- oring is basedColoring on lithology is based on (black: lithology basalt, (black: g basalt,reen: green:trachybasalt, trachybasalt, yellow: yellow: trachyandesite, trachyandesite, red: red: trachyte). trachyte). Finally, regression models have been calculated for the morphometric parameters and Finally,ages of the regression cones (Figure models 17). Both have the beenHco/W calculatedco and (DTM-based) for the averagemorphometric slope values parameters and agesshow of relatively the cones large (Figure scattering 17 in). B theoth figure the showingHco/Wco and a certain (DTM trend.-based) The simplest average linear slope values showmodels relatively are very large similar scattering in the case in ofthe regression figure showing of Hco/Wco avalues. certain This trend. is due The to the simplest age linear grouping of two typical age ranges: 10–14 ka and 40–47 ka. In this case, cones of various modelslithologies are very are similar in-line with in the the regressioncase of regression models. of Hco/Wco values. This is due to the age groupingConsidering of two typical the (DTM-based) age ranges: average 10–14 slope ka anglesand 40 (Figure–47 k 17a.), In the this situation case is, cones slightly of various lithologiesdifferent: are ordinary in-line leastwith squares the regression fit and Theil–Sen models. estimator practically result in being the same, whereas RANSAC finds a rather different model. Only a few cones are aligned with the former two models (mostly, but not exclusively basalts); RANSAC inliers are more numerous and all types of lithologies are represented in balanced numbers. Assuming limited validity to the regression models, either ca. 0.13 degrees/ky or 0.32 degrees/ky edifice destruction would be an estimate on destructional rates. Since these values have very high estimation error, at the current level of knowledge, they cannot be considered as of erosional origin. In order to get better models, further morphometric studies and more complex geomorphometric approaches are needed. The Chaîne des Puys is a young volcanic field, compared with many studies, which range into millions of years, so it is likely that the degradational processes are more difficult to detect at such short time scales. In addition, as the cones are closely spaced, they tend to erupt material onto one another, and thus there is a possible constructional effect from material addition from nearby cones after an eruption is finished. This is something that may occur in all volcanic fields but would be most important where the cones are close, like in the Chaîne des Puys.

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Figure 17. (a) regression models for age vs. Hco/Wco ratio. All three models are similar with many outliers; (b) regression Figure 17. (a) regression models for age vs. Hco/Wco ratio. All three models are similar with many outliers; (b) regression models for age vs. (DTM-based) average slope. RANSAC results in a completely different model, whereas the results of models for age vs. (DTM-based) average slope. RANSAC results in a completely different model, whereas the results of ordinary least squares fit and Theil–Sen estimator are similar. ordinary least squares fit and Theil–Sen estimator are similar. 5. Conclusions Considering the (DTM-based) average slope angles (Figure 17), the situation is slightlyFrom different: the results ordinar of they morphometricleast squares fit analysis and Theil of the–Sen 25 cones estimator studied practically of the Chaîne result in des Puys, the following conclusions can be drawn: being the same, whereas RANSAC finds a rather different model. Only a few cones are aligned1. The with general the former assumption two models for scoria (mostly, cones that but the not basal exclusively width (Wco) basalts); correlates RANSAC with the inliers height of the cone (Hco) could be verified. However, Hasenaka & Carmichael’s [35] are more numerous and all types of lithologies are represented in balanced numbers. As- calculation of average slopes overestimates the real slope values, sometimes to a great sumingextent. limited We validity suggest thatto the this regression method should models, be replaced either by ca. DTM-derived 0.13 degrees/ky slope values, or 0.32 degrees/kyin edifice agreement destruction with a number would of subsequentbe an estimate studies on (e.g.,destructional [14]). rates. Since these val- ues2. haveIn addition,very high a moreestimation detailed error, morphometric at the current assessment level that of tacklesknowledge the circular, they sym-cannot be consideredmetry as of of the erosional cones reveals origin. that specific In order groups to get behave better differently, models, and,further in some morphometric cases, studiesthey and definemore complex a separate geomorphometric trend or no trend canapproaches be detected. are needed. The slope distributions Theextracted Chaîne from des LiDARPuys is digital a young terrain volcanic models field, indicate compared greater with variability many with studies, lithol- which range intoogy/composition millions of years, than withso it age,is likely at least that over the the degradational age range of the processes volcanoes are here. more difficult to detect at such short time scales. In addition, as the cones are closely spaced, they tend to erupt material onto one another, and thus there is a possible constructional effect from material addition from nearby cones after an eruption is finished. This is something that may occur in all volcanic fields but would be most important where the cones are close, like in the Chaîne des Puys.

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3. Nevertheless, within the same lithological group, subtle but possibly systematic trends can be detected for decreasing morphometric values (e.g., slope) with the age. Cones of different lithology/composition can have different relationships, and thus different composition cones produce slightly different shapes. 4. Despite the attempt to characterize them separately, we can conclude that the time span of the trachytic cones/domes of Chaîne des Puys is too short for significant differences or correlations with the age to be detected with confidence. 5. Morphometric sectorization resulted in separation into various types of symmetries. Some cones are close to a regular shape, but the majority of the cones are not circularly symmetric. These cones, however, often show other types of symmetry. The radar diagrams of specifically processed slope distributions show similarities and dissimilarities. These observations are encouraging to perform further statistical analysis of sectorial data that might reveal further relationships with tectonics, slope or wind directions. 6. The initial aim to explore relationships of age and lithology with and morphometric parameters showed that lithology is a strong control, but only a faint age effect was detectable over the timespan of just under 100,000 years. The variability of original cone morphometry led to large errors in estimated rates. More detailed geomorphological approaches integrating lithological factors would be a next step, and applying the methods to a longer time span. 7. The variability depicted in the morphometry is connected largely to the lithology and thus eruptive processes, and/or potentially with the angle of repose of various types of scoria. The methods here have the potential to explore such processes over a volcanic field. Further work is needed to understand all the diverse parameters, especially how different compositions produce different shapes, and how symmetry is connected to different factors.

Author Contributions: Conceptualization, B.S. and F.V.; methodology, B.S.; software, B.S. and F.V.; validation, B.v.W.d.V. and D.K.; investigation, ALL; resources, B.v.W.d.V.; writing—original draft preparation, ALL; writing—review and editing, B.v.W.d.V. and D.K.; visualization, F.V. and B.S. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available upon request from the corresponding author. Acknowledgments: The authors are deeply indebted to the three anonymous reviewers whose constructive comments, linguistic, and structural suggestions considerably improved our manuscript. LIDAR DTM CRAIG open data site (www.craig.fr; accessed on 29 April 2021). For the statistical evaluations and visualization, various functions of the matplotlib and scikit-learn Python packages have been used, the authors’ work (e.g., Dan Nelson’s and Florian Wilhelm’s) is acknowledged. Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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