Marsden, LH

Neuberg, JW

Thomas, ME

English

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CORRECTIONpublished: 03 March 2020doi: 10.3389/feart.2020.00046Frontiers in Earth Science | www.frontiersin.org 1 March 2020 | Volume 8 | Article 46Edited and reviewed by:Yosuke Aoki,The University of Tokyo, Japan*Correspondence:Luke H. Marsdeneelhm@leeds.ac.ukSpecialty section:This article was submitted toVolcanology,a section of the journalFrontiers in Earth ScienceReceived: 31 January 2020Accepted: 12 February 2020Published: 03 March 2020Citation:Marsden LH, Neuberg JW andThomas ME (2020) Corrigendum:Topography and Tilt at Volcanoes.Front. Earth Sci. 8:46.doi: 10.3389/feart.2020.00046Corrigendum: Topography and Tilt atVolcanoesLuke H. Marsden*, Jürgen W. Neuberg and Mark E. ThomasSchool of Earth & Environment, The University of Leeds, Leeds, United KingdomKeywords: topography, tilt, shear stress, pressure, numerical modeling, volcanoA Corrigendum onTopography and Tilt at Volcanoesby Marsden, L. H., Neuberg, J. W., and Thomas, M. E. (2019). Front. Earth Sci. 7:317.doi: 10.3389/feart.2019.00317In the original article, there was an oversight, resulting in the previous work of Beauducel andCarbone (2015) and Albino et al. (2011) not being cited.A correction has been made to Section 2. Influence of the Original Slope Angle, paragraph 3.Inflation or deflation of a magma reservoir is commonly modeled analytically using a pointsource (Mogi, 1958), where the orientation of the displacement field at the surface varies withhorizontal distance from the source. The point source approximation is valid for a spherical sourcewhere the source radius is small with respect to the distance from that source. Addressing theinfluence of topography, Cayol and Cornet (1998) showed that displacement produced by reservoirpressure is greater at lower elevation, where the surface is closer to the source. Delving further,Ronchin et al. (2015) found that the slope angle θ is a secondary control on the displacement.Through numerical modeling, they showed that the maximum displacement is produced wherethe surface is perpendicular to a line between the surface and source, where the exposure to thepressure source is greatest. We refer to the slope angle with the maximum exposure to the sourceat any point as θmaxexp, and the deviation from this angle as θ − θmaxexp. Hence, tilt generated bya point or spherical source is influenced by the slope angle. We vary this over a 90◦ range arounda point centered at x = 3 km, z = 3 km (Figure 1C). We calculate u and w at each point along theslope using it’s distance from the centre of a Mogi source (x = 0 km, z =−1 km)uvw = α31P1− νEmx/R3y/R3d/R3 , (7)where 1P is the pressure, α is the source radius, d is the vertical distance between each point andthe source center, ν is the Poisson’s ratio and Em is the Young’s modulus. v is the displacement inthe y direction, which is not used in this 2D example, but is considered in the 3D modeling later insection 5. The new slope angle can be computed as the derivative of the new coordinates for eachpoint along the slope, x + u and z + w. The tilt 1θ is then calculated as the change in slope angle.Beauducel and Carbone (2015) suggested that tilt induced by a spherical pressure source is greaterwhere the slope of the edifice is steeper. However, we find that the tilt 1θ/1θmax is greatest wherethe surface is 45◦ from the angle of maximum exposure, i.e., where θ − θmaxexp = 45◦ (Figure 1C).In other words, no tilt is produced where the surface is parallel or perpendicular to the displacementfield, and the highest tilt is generated where the surface is 45◦ from this. This, unsurprisingly, is aspreviously shown for a purely horizontal displacement field.Marsden et al. Corrigendum: Topography and Tilt at VolcanoesA correction has been made to Section 5. Real Topography,paragraphs 1 and 2.The 2D axisymmetric models in sections 3, 4 are usefulin demonstrating how exaggerated features of simplifiedtopography can affect the tilt. However, it is important toconsider whether the effect of real topography is significant ornegligible. To do this, we present results of 3D deformationmodeling, using 10m × 10m digital elevation models (DEMs)of Tungurahua volcano, Ecuador and Soufrire Hills volcano,Montserrat (SHV), and compare how tilt varies spatially acrossboth DEMs where topography is the only variable. We againasses the topographic effect on tilt induced by each of the sourcesdescribed in section 1. The mesh must be suitably fine suchthat the resolution of the DEM is matched. Beauducel andCarbone (2015) investigated the influence of real topographyon tilt produced by a spherical pressure source. However, sincethey use a spatially variant mesh with a minimum elementsize of 200m, they thus do not consider the influence of localtopography, which we show can significantly influence the tilt(section 4). Here, we use spatially variant triangular mesh acrossthe entire DEM, to a distance of almost 8 km in x and y fromthe conduit, with a minimum element size of ∼1m wherethe topography is most complex. The model is extrapolated todistance of 40 km in x and y and a depth of 50 km, to ensure thatthe deformation within the region that the DEM covers is notaffected by the boundary constraints, described in section 3. Amuch coarser mesh (maximum element size ∼7 km) is used forthis extrapolation. We consider the maximum tilt for 1θ , whichis always positive and not necessarily radial to the source.In defining the reference tilt, 1θr, we followed the commonpractice where topography is not considered. Therefore, 1θAMPis the factor by which the modeled tilt is scaled when topographyis considered relative to the common practice when it is not. Fortilt generated by reservoir pressure, an analytical solution wasused (Mogi, 1958) (Equation 7), taking the x, y, z coordinatesof each individual point across the mesh to calculate 1θr. Fortilt produced by conduit pressure or shear stress, in the absenceof an available analytical solution, we obtain 1θr from 2Daxisymmetricmodels using a constant slope, based on the averagedip angle radially away from the summit for each edifice. Thismimics the approach of Albino et al. (2011), who representtopography using only this first-order approximation. Thecomputation of this average dip angle, and a full description ofthese 2D axisymmetric models, is included in the SupplementaryMaterial, Section S1.The authors apologize for this oversight and state that thisdoes not change the scientific conclusions of the article in anyway. The original article has been updated.REFERENCESAlbino, F., Pinel, V., Massol, H., and Collombet, M. (2011). Conditionsfor detection of ground deformation induced by conduit flow andevolution. J. Geophys. Res. Solid Earth 116:B06201. doi: 10.1029/2010JB007871Beauducel, F., and Carbone, D. (2015). A strategy to explore the topography-driven distortions in the tilt field induced by a spherical pressure source:the case of Mt Etna. Geophys. J. Int. 201, 1471–1481. doi: 10.1093/gji/ggv076Cayol, V., and Cornet, F. H. (1998). Effects of topography on theinterpretation of the deformation field of prominent volcanoes—application to etna. Geophys. Res. Lett. 25, 1979–1982. doi: 10.1029/98GL51512Mogi, K. (1958). Relations between the eruptions of various volcanoes and thedeformations of the ground surfaces around them. Bull. Earthq. Res. Inst. Univ.Tokyo 36, 99–134.Ronchin, E., Geyer, A., and Martí, J. (2015). Evaluating topographic effects onground deformation: insights from finite element modeling. Surv. Geophys. 36,513–548. doi: 10.1007/s10712-015-9325-3Copyright © 2020 Marsden, Neuberg and Thomas. This is an open-access articledistributed under the terms of the Creative Commons Attribution License (CC BY).The use, distribution or reproduction in other forums is permitted, provided theoriginal author(s) and the copyright owner(s) are credited and that the originalpublication in this journal is cited, in accordance with accepted academic practice.No use, distribution or reproduction is permitted which does not comply with theseterms.Frontiers in Earth Science | www.frontiersin.org 2 March 2020 | Volume 8 | Article 46

Corrigendum: Topography and Tilt at Volcanoes

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Corrigendum: Topography and Tilt at Volcanoes

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