Surface fitting in geomorphology — Examples for regular-shaped volcanic landforms

Abstract

In nature. several types of landforms have simple shapes: as they evolve they tend to take on an ideal, simple geometric form such as a cone. an eIlipsoid or a paraboloid. VoIcanic landforms are possibly the best examples of this "ideal" geometry. since they develop as regular surface features due to the point-Iike (circular) or fissure-Iike (linear) manifestation of voIcanic activity. In this papero we present a geomorphometric method of fitting the "ideal" surface onto the real surface of regularshaped volcanoes through a number of case studies (Mt. Mayon. Mt. Somma. Mt. Semeru. and Mt. Cameroon). VoIcanoes with circular. as weIl as eIliptical, symmetry are addressed. For the best surface fit. we use the minimization Iibrary MINUIT which is made freely available by the CERN (European Organization for Nuclear Research). This Iibrary enables us to handle aIl the available surface data (every point of the digital elevation model) in a onestep. half-automated way regardless of the size of the dataset. and to consider simultaneously aIl the relevant parameters of the selected problem. such as the position of the center of the edifice. apex height. and cone slope. thanks to the highly performing adopted procedure. Fitting the geometric surface. along with calculating the related error. demonstrates the twofold advantage ofthe method. FirstIy. we can determine quantitatively to what extent a given voIcanic landform is regular. Le. how much it foIlows an expected regular shape. Deviations from the ideal shape due to degradation (e.g. sector coIlapse and normal erosion) can be used in erosion rate caIculations. Secondly. ifwe have a degraded volcanic landform. whose geometry is not c1ear. this method of surface fitting reconstructs the original shape with the maximum precision. Obviously. in addition to voIcanic landforms. this method is also capable of constraining the shapes of other regular surface features such as aeolian. glacial or periglacial landforms.

Construction and destruction rates of volcanoes within tropical environment: Examples from the Basse-Terre Island (Guadeloupe, Lesser Antilles)

“…Lavigne et al, 2013), which is an improvement of a previous reconstruction method . It now considers the erosion having affected the present ridges and finds, using the simplex algorithm, the location of axis of revolution, the elliptical shape and the average profile that defines the altitudes (Z SV ) of the lowest possible surface of revolution passing above the constraining points (Favalli et al, 2014).…”

Initial shape reconstruction of a volcanic island as a tool for quantifying long-term coastal erosion: the case of Corvo Island (Azores)

“…One solution is to calculate the barycenter of the concentric elevation contours, but this method is only applicable for very regular edifices that have experienced very little erosion (Karátson et al, 2010). Instead, we have chosen here a second method developed by Favalli et al (2014) to determine the center of Mount Somma (Italy), a highly eroded edifice of which only the northwestern part of the slopes remains. This method determines the position of the center of the edifice as the center point of the radial drainage network (Fig.…”

“…2). Following the work of Karátson et al (2010Karátson et al ( , 2016 and Favalli et al (2014), we fit the aerial topographic data with a decreasing exponential because it is best suited for the type of volcano we are in-terested in, i.e., initially radially symmetric stratovolcanoes (Favalli et al, 2014;Karátson et al, 2016). The exponential profile was originally noted by Gee et al (2001) but without a theoretical explanation.…”

“…In addition, its end is relatively predictable, with the disappearance of the island under subsidence or erosion (Ramalho et al, 2013). It is also plausible to reconstruct the paleotopography of volcanoes (Karátson et al, 2010;Favalli et al, 2014;Karátson et al, 2016).…”

The Characteristics of Volcanic Eruption in Indonesia

“…Semeru is one of the most active volcanoes in East Java. This mountain is better known as the peak of Mahameru because it has highest mountains on the Java island [10]. Semeru is located on 08′06.5′ South latitude and 112°55′ East longitude.…”