# Morphometric analysis and classification of the three-dimensional geometry of shatter cones

David Baratoux, S. Bouley, W. U. Reimold, L Baratoux

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David Baratoux, S. Bouley, W. U. Reimold, L Baratoux. Morphometric analysis and classification of the three-dimensional geometry of shatter cones. Meteoritics and Planetary Science, 2016, 51 (8):), pp.1460-1476 (IF 2,707). 10.1111/maps.12610 . hal-01415962

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# Morphometric analysis and classification of the three-dimensional geometry of shatter cones

David BARATOUX<sup>1,2*</sup>, Sylvain BOULEY<sup>3,4</sup>, Wolf Uwe REIMOLD<sup>5,6</sup>, and Lenka BARATOUX<sup>1,2</sup>

*(Received 15 July 2015; revision accepted 01 December 2015)*

**Abstract**—Shatter cones are curved fractures decorated with divergent striations that are exclusively associated with impact metamorphism. The terminology “cone” was chosen from the observation of complete or near-complete, roundish, axisymmetric objects with a well-defined apex/apical area. It also imprecisely extends to the curved or subplanar fractures decorated with striations commonly observed at impact sites. The geometry of these objects is therefore highly variable but its significance was never addressed due to the lack of appropriate data. Here, we apply two methods to derive shape models of shatter cones. The first one is based on images acquired using a commercial camera and may be applied on large samples in the field. The second one uses an articulated arm equipped with a digital laser scanner and produces high-resolution and precision shape models of hand-sized samples. The analysis of 20 shape models of shatter cones from nine different impact sites indicates that the surface of shatter cones may be described by quadric surfaces and are generally consistent with hyperboloids, whereas occurrence of paraboloid objects cannot be ruled out. The surface characteristics are generally not consistent with the mathematical definition of a cone. The value of these shape models to discriminate between the different hypotheses of formation of shatter is still limited, as it remains to be resolved which type of surface pertains to which hypothesis. This requires theoretical developments, and experimental or numerical simulations of the propagation of tensile fractures associated with shock waves.

## INTRODUCTION

Shatter cones are represented by a range of curved fractures decorated with divergent striations and occurring exclusively in the context of impact structures. The term originated from the spectacular samples of complete and isolated cones first described from the Steinheim Basin impact structure (Branco and Fraas 1905). However, such specimens are relatively rare for

most other impact structures, and the term loosely applies to a wide range of curviplanar fractures and composite objects composed of several curved fractures associated with the typical divergent striations (e.g., Ferrière and Osinski 2012).

In geometry, a conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point—the apex or vertex—and any point of some fixed space curve—the directrix—that does

<footnote n="1"><sup>1</sup>Géosciences Environnement Toulouse, University of Toulouse, CNRS & IRD, 31400 Toulouse, France</footnote>

<footnote n="2"><sup>2</sup>Institut Fondamental d'Afrique Noire, Univesité Cheikh Anta Diop, Dakar, Sénégal</footnote>

<footnote n="3"><sup>3</sup>GEOPS—Géosciences Paris Sud, Univ. Paris-Sud, CNRS, Université Paris-Saclay, Rue du Belvédère, Bât. 504-509 91405 Orsay, France</footnote>

<footnote n="4"><sup>4</sup>Institut de Mécanique Céleste et de Calcul des Ephémérides, UMR8028, 77 Avenue Denfert Rochereau, 75014 Paris, France</footnote>

<footnote n="5"><sup>5</sup>Museum für Naturkunde—Leibniz Institute for Evolution and Biodiversity Science, Invalidenstrasse 43, 10115 Berlin, Germany</footnote>

<footnote n="6"><sup>6</sup>Humboldt Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany</footnote>

<footnote>*Corresponding author. E-mail: david.baratoux@get OBS-MIP.fr</footnote>
not contain the apex. Each of those lines is called a generatrix of the surface. If the directrix is a circle, the cone is said to be a circular cone. If the directrix is an ellipse, the cone is said to be an elliptical cone. Even if the term shatter cone is retained for historical reasons, it has been already noted that topographic profiles originating at the apex are curved (Sagy et al. 2004) and a more general algebraic expression would be needed to describe the shape of these fractures. However, the type of surface associated with shatter cones has never been addressed due to the lack of appropriate data. Here, we propose that shatter cones may be adequately described by quadric surfaces. Quadric surfaces are relatively common in nature and are defined in Euclidian space as the zeroes of a quadratic polynomial. Such mathematical objects include ellipsoids, hyperboloids, and paraboloids. Cylinders, cones, and planes are particular (degenerate) cases of quadric surfaces. The two-dimensional (2-D) topographic profiles recorded by Sagy et al. (2004) do not allow us to assess the three-dimensional (3-D) geometry of these objects. 3-D shape models of experimentally produced tiny shatter cones (a few millimeters in size) have been recently obtained in the context of a MEMIN experiment (Wilk and Kenkmann 2015a, 2015b). The analysis of the surface morphology of the tiny sample was carried out with a Bruker AXS Contour GT-K0 white light interferometer (WLI). The study of Wilk and Kenkmann (2015a) focused on the morphology of the striations, and on the presence of vesicular melt films but did not address the macroscopic shape of the shatter cones produced in these experiments.

Therefore, the first objective of this paper is to determine whether shatter cones may be described by one or several types of quadric surfaces using a database of shape models generated for selected shatter cones from various impact sites and for different types of host rock. A second objective is to discuss whether and how this 3-D geometric information can be used to evaluate the different hypotheses for the formation of shatter cones and the potential of this approach to settle a debate about this fundamental question in impact science that started about 50 years ago. In the first section, we describe the two methods used for the generation of shape models. The first method may be directly applied in the field using a commercial digital camera and is useful to produce shape models of voluminous objects that would be hardly transported for analysis in the laboratory. The second method uses the laser-scanning technique and produces high-resolution and high-precision models. It requires the transport of samples back to the laboratory (or alternatively the transport of the digital laser scanner and arm into the field, which is generally more expensive). The second section describes how shape

models may be classified to determine the type of quadric surface that corresponds to each sample. Twenty shape models for samples from nine different impact sites were produced and are presented in the third section with the characteristics of the inferred quadric surfaces. Analyses of quadric types versus models of shatter cone formation and suggestions for future work are presented in the Discussion section.

# ACQUISITION PROCEDURE OF DIGITAL SHAPE MODELS

## 3-D Field Modeling Using Digital Cameras

Shape models may be produced using a stack of digital images processed although the Helicon Focus software (http://www.heliconsoft.com/). The Helicon Focus software is a proprietary commercial digital image-processing tool that is primarily used to increase the depth of field (DOF) in macrophotography, where the DOF is typically very small. To increase DOF, Helicon Focus merges differently focused images together to create an image where the subject is entirely in focus. In addition, the stacking results may be combined into a qualitative 3-D view of the object. The 3-D information may be exported in various digital formats but is not calibrated. The transformation equation of 3-D outputs into *x*,*y*,*z* coordinates depends on the conditions of acquisition including the characteristics of the optical system, the distance to the object, the number of images used in the stack, and the step in the position of the focal plane between each image. An empirical approach was developed in order to determine a calibration curve valid for only a given set of images having similar conditions of acquisition. In practice, this corresponds to a single sample and a single stack of images, as the conditions of acquisition in the field are different for each sample. For this purpose, we have placed within the field of view of each picture a spherical object (named here the “reference sphere”) with colorful textures on its surface. The reference sphere is actually a popular toy rubber-bouncing ball with painted motives on its surface (Fig. 1a). The texture facilitates the image processing and 3-D reconstruction, as Helicon Focus uses the textural information to determine where an object is in focus (it cannot produce 3-D information from an object uniform in color). The reference plane (*z* = 0) of the shape model contains the center of the reference sphere and is perpendicular to the line of sight. Pixel positions are converted into meters using scale factors obtained from the (*x*<sub>pix</sub>, *y*<sub>pix</sub>) pixel coordinates of the circle at the intersection of the sphere with the reference plane. Apparent elevations provided by Helicon Focus in the *z*-direction (*z*<sub>app</sub>) are converted into elevations in meters
<figure><img src="image_2.png" /><figcaption>Fig. 1. Left: Toy rubber painted ball, 6 cm in diameter, used for the calibration of shape models produced by Helicon Focus, example of use in the field at Gosses Bluff, Australia. Right: Digital laser scanner with 7-axis articulated arm (MCA II 7 axis ®Nikon).</figcaption></figure>

using the known surface coordinates of the reference sphere (x<sub>ref</sub>, y<sub>ref</sub>, z<sub>ref</sub>). A linear or second-order polynomial calibration curve is derived from least-squares adjustment of z<sub>app</sub> to z<sub>ref</sub>. The calibration coefficient and scale factors determined for each series (stack) of images are then applied to produce a shape model comprising a list of x,y,z coordinates in meters. The cloud of points is then interpolated into a regular grid. The resolution of the grid is chosen to be comparable to the average distance between the points in the cloud.

## 3-D Laboratory Laser Scanning

Shape models are produced using a digital laser scanner (MCA II 7 axis ®Nikon, Fig. 1b) that can be moved around the sample with an articulated arm. The articulated arm has seven axes of rotation allowing us to explore the geometry of a sample in any direction. A cloud of points with (x,y,z) coordinates (in the reference frame of the scanning arm) is generated during the scanning procedure. A subset of the cloud of points is then selected to eliminate any measurements from the background (e.g., the table on which the sample rests during the acquisition procedure), and to remove faces of the samples that are normal fractures or joints, which are not decorated by striations). A final selection of a subset of points ensures that the surface may be described using a Cartesian equation z = f(x,y) in an appropriate reference frame to facilitate data interpolation, manipulation, and visualization using standard software products and formats for handling and sharing of Digital Elevation Models in geosciences. A major inconvenience of this choice is that we were unable to consider the entire 3-D information for

complete cones, or near-complete cones, and only a portion of them (e.g., half of a complete cone at best) will be considered in the surface classification process. Then, a reference plane is determined by least-squares adjustment of the subset of the cloud of points. The normal vector to this plane defines the z-axis of the new reference frame. The coordinates of the cloud of points are rotated into the new reference frame. The acquisition procedure generates a cloud of points that is heterogeneous in density with extremely variable redundancy (depending on the time spent on a given part of the sample). The cloud of points is therefore interpolated into a regular grid using 2-D Delaunay triangulation and linear interpolation, using existing routines built in the IDL® language. The resolution of the grid is chosen to be comparable with the average distance between the points in the cloud. The data set may then be manipulated as a standard Digital Elevation Model, characterized by its resolution (pixel size) and vertical precision (precision along the z-axis).

## CLASSIFICATION OF DIGITAL SHAPE MODELS

### Quadric Surfaces—Definition and Classification Procedure

The general equation of a quadric surface in a Cartesian frame is given by a quadratic polynomial expression:

$$l_1x^2 + l_2y^2 + l_3z^2 + l_4xy + l_5yz + l_6xz + l_7x + l_8y + l_9z + l_{10} = 0 \quad (1)$$
<table><caption>Table 1. Standard formulas of quadric surfaces with the values of D and Δ, and corresponding expressions for Q, g, and d (see text for definitions).</caption><thead><tr><th>Surface type</th><th>Standard equation</th><th>Q</th><th>g</th><th>d</th></tr></thead><tbody><tr><td>Single-sheet hyperboloid Δ > 0, D ≠ 0</td><td>x<sup>2</sup>/a<sup>2</sup> + y<sup>2</sup>/b<sup>2</sup> - z<sup>2</sup>/c<sup>2</sup> = 1</td><td>1/a<sup>2</sup><br>1/b<sup>2</sup><br>-1/c<sup>2</sup></td><td>0<br>0<br>0</td><td>1</td></tr><tr><td>Dual-sheet hyperboloid Δ < 0, D ≠ 0</td><td>x<sup>2</sup>/a<sup>2</sup> + y<sup>2</sup>/b<sup>2</sup> - z<sup>2</sup>/c<sup>2</sup> = -1</td><td>1/a<sup>2</sup><br>1/b<sup>2</sup><br>-1/c<sup>2</sup></td><td>0<br>0</td><td>-1</td></tr><tr><td>Ellipsoid Δ < 0, D ≠ 0</td><td>x<sup>2</sup>/a<sup>2</sup> + y<sup>2</sup>/b<sup>2</sup> + z<sup>2</sup>/c<sup>2</sup> = 1</td><td>1/a<sup>2</sup><br>1/b<sup>2</sup><br>1/c<sup>2</sup></td><td>0<br>0<br>0</td><td>1</td></tr><tr><td>Elliptical cone Δ = 0, D ≠ 0</td><td>x<sup>2</sup>/a<sup>2</sup> + y<sup>2</sup>/b<sup>2</sup> - z<sup>2</sup>/c<sup>2</sup> = 0</td><td>1/a<sup>2</sup><br>1/b<sup>2</sup><br>-1/c<sup>2</sup></td><td>0<br>0<br>0</td><td>1</td></tr><tr><td>Hyperbolic paraboloid Δ > 0, D = 0</td><td>x<sup>2</sup>/a<sup>2</sup> - y<sup>2</sup>/b<sup>2</sup> - z = 0</td><td>1/a<sup>2</sup><br>-1/b<sup>2</sup><br>0</td><td>0<br>0<br>-1</td><td>0</td></tr><tr><td>Elliptical paraboloid Δ < 0, D = 0</td><td>x<sup>2</sup>/a<sup>2</sup> + y<sup>2</sup>/b<sup>2</sup> - z = 0</td><td>1/a<sup>2</sup><br>1/b<sup>2</sup><br>0</td><td>0<br>0<br>-1</td><td>0</td></tr><tr><td>Elliptical cylinder Δ = 0, D = 0</td><td>x<sup>2</sup>/a<sup>2</sup> + y<sup>2</sup>/b<sup>2</sup> = 1</td><td>1/a<sup>2</sup><br>1/b<sup>2</sup><br>0</td><td>0<br>0<br>0</td><td>1</td></tr><tr><td>Hyperbolic cylinder Δ = 0, D = 0</td><td>x<sup>2</sup>/a<sup>2</sup> - y<sup>2</sup>/b<sup>2</sup> = 1</td><td>1/a<sup>2</sup><br>-1/b<sup>2</sup><br>0</td><td>0<br>0<br>0</td><td>1</td></tr><tr><td>Parabolic cylinder Δ = 0, D = 0</td><td>x<sup>2</sup> + 2ay = 0</td><td>1<br>0<br>0</td><td>0<br>2a<br>0</td><td>0</td></tr><tr><td>Plane Δ = 0, D = 0</td><td>az + by + cx = 1</td><td>0<br>0<br>0</td><td>a<br>b<br>c</td><td>1</td></tr></tbody></table>

By making a suitable rotation and translation, the quadric surfaces may be put into one of the standard forms corresponding to different types of quadric surfaces including ellipsoid, elliptical paraboloid, hyperbolic paraboloid, elliptic hyperboloid of one

sheet, elliptical hyperboloid of two sheets, elliptical cone, elliptical cylinder, hyperbolic cylinder, and parabolic cylinder (Table 1). These standard forms may be expressed using the following algebraic relationship:
$$x^T Q x + g x - d = 0 \qquad (2)$$

where **x** is the vector of Cartesian coordinates, **g** is a vector, **d** a scalar value, and **Q** is a diagonal matrix. The corresponding values of the diagonal terms of **Q**, the values of **g**, and **d** are given in Table 1. The different types of surfaces may be classified as a function of the two following invariants **D** and **Δ** (Dai et al. 2007):

$$D = \begin{vmatrix} l_1 & l_4/2 & l_6/2 \\ l_4/2 & l_2 & l_5/2 \\ l_6/2 & l_5/2 & l_3 \end{vmatrix} \qquad (3)$$

$$\Delta = \begin{vmatrix} l_1 & l_4 & l_6/2 & l_7/2 \\ l_4/2 & l_2 & l_5/2 & l_8/2 \\ l_6/2 & l_5/2 & l_3 & l_9/2 \\ l_7/2 & l_8/2 & l_9/2 & l_{10} \end{vmatrix} \qquad (4)$$

Surfaces, for which **D** equals 0, form the group of noncentral quadric surfaces (hyperbolic paraboloids, elliptical paraboloids, cylinder, and plane). Central quadric surfaces include the single-sheet hyperboloid, the dual-sheet hyperboloid, the ellipsoid, and the cone. Any quadric surface may be expressed by one of these equations after an appropriate rotation and translation is achieved.

## Classification of Digital Shape Models

The determination of the parameters of a given standard form from a shape model would be a nonlinear inverse problem with up to nine unknowns: six unknowns for translation and rotation and up to three unknowns for the quadric parameters in standard form. In principle, nonlinear inversion would be a possible approach, but it would require intensive computer calculation given the resolution and associated large number of points in shape models obtained by laser scanning. Convergence problems may also occur, and an alternative approach based on the general quadric equation in Cartesian space (Equation 1) may be implemented (Hall et al. 1982; Dai et al. 2007). Assuming $l_{10}$ is non-zero in Equation 1, the general quadric equation may be written with nine independent coefficients:

$$k_1 x^2 + k_2 y^2 + k_3 z^2 + k_4 xy + k_5 yz + k_6 xz + k_7 x + k_8 y + k_9 z = 1 \qquad (5)$$

We summarize here the procedure given in Dai et al. (2007). The reader is referred to the original paper

for mathematical details. The determination of the nine independent coefficients from a shape model for a given surface in Cartesian coordinates ($z_i = f(x_i, y_i)$) is a straightforward linear inverse problem and is solved by a least-squares adjustment that minimizes the following expression:

$$e = \| \mathbf{A} \mathbf{k} - \mathbf{1} \|^2 \qquad (6)$$

where **k** is the vector of the $k_1-k_9$ coefficients, **1** is a n-component vector, and **A** is the following ($n \times 9$) matrix, where **n** is the number of points in the shape model:

$$A = \begin{pmatrix} x_1^2 & y_1^2 & z_1^2 & x_1 y_1 & y_1 z_1 & x_1 z_1 & x_1 & y_1 & z_1 \\ x_2^2 & y_2^2 & z_2^2 & x_2 y_2 & y_2 z_2 & x_2 z_2 & x_2 & y_2 & z_2 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ x_i^2 & y_i^2 & z_i^2 & x_i y_i & y_i z_i & x_i z_i & x_i & y_i & z_i \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ x_n^2 & y_n^2 & z_n^2 & x_n y_n & y_n z_n & x_n z_n & x_n & y_n & z_n \end{pmatrix} \qquad (7)$$

The solution **K** is then given using the equation:

$$\mathbf{k} = (\mathbf{A}^T \mathbf{A})^{-1} \mathbf{A}^T \mathbf{1} \qquad (8)$$

The equation is solved numerically using standard matrix inversion or QR decomposition, if the standard inversion procedure fails. Once the **k** coefficients have been estimated, the invariants **D** and **Δ** are evaluated and the surface may be classified as a function of the values of these invariants. Ellipsoid and dual-sheet hyperboloid have both non-null values for **D** and **Δ** negative, and must be distinguished from the estimation of the matrix **A** from the **k** coefficients using the procedure described in Dai et al. (2007). The type of the quadric surface may be determined from the sign of the third term of the diagonal terms of **Q** (see Table 1). The quality of the fit is given by the following expression:

$$\sigma = \sqrt{\frac{e}{n}} \qquad (9)$$

where **e** is given in Equation 6 and **n** is the number of points in the shape model.

It is of note that a quadric with an arbitrary small but nonzero value of **D** may be classified as a central quadric, and it would be somewhat arbitrary to fix a minimal value of **D** for classification as a central quadric. Therefore, even if the surface is classified as a central quadric it is important to check whether a noncentral
<table><caption>Table 2. List of the shape models for shatter cones from different impact sites.</caption><thead><tr><th>Crater name</th><th>Longitude</th><th>Latitude</th><th>Diameter (km)</th><th>Lithology</th><th>Helicon Focus models</th><th>Laser shape models</th></tr></thead><tbody><tr><td>Gosses Bluff</td><td>132°19′E</td><td>23°49′S</td><td>22</td><td>Sandstone</td><td>4</td><td>0</td></tr><tr><td>Haughton</td><td>89°41′W</td><td>75°22′N</td><td>23</td><td>Dolomite</td><td>1</td><td>1</td></tr><tr><td>Jebel Waqf as Suwwan</td><td>36°48′E</td><td>31°03′N</td><td>5.5</td><td>Sandstone</td><td>1</td><td>1</td></tr><tr><td>Serra da Cangalha</td><td>46°52′W</td><td>8°5′S</td><td>12</td><td>Sandstone</td><td>0</td><td>1</td></tr><tr><td>Rochechouart</td><td>0°56′E</td><td>45°50′N</td><td>23</td><td>Granite</td><td>0</td><td>2</td></tr><tr><td>Steinheim Basin</td><td>10°4′E</td><td>48°41′N</td><td>3.8</td><td>Limestone</td><td>0</td><td>3</td></tr><tr><td>Vargeão Dome</td><td>52°07′W</td><td>26°50′S</td><td>12</td><td>Basalt</td><td>0</td><td>1</td></tr><tr><td>Vista Alegre</td><td>52°41′W</td><td>25°57′S</td><td>9.5</td><td>Basalt</td><td>0</td><td>1</td></tr><tr><td>Vredefort</td><td>27°30′E</td><td>27°0′S</td><td>160</td><td>Shale</td><td>0</td><td>4</td></tr><tr><td>Total no. of shape models</td><td></td><td></td><td></td><td></td><td>6</td><td>14</td></tr></tbody></table>

quadric surface would produce an equally good fit. Such ambiguous cases would result in similar values of $\sigma$ for a central and a noncentral quadric surface. In order to determine $\sigma_{ncq}$ (for noncentral quadric surface), an inversion where coefficients $k_1-k_9$ are forced to satisfy the equation $D = 0$ is also achieved. Incorporation of an additional equation implies that the inverse problem has now eight unknowns and one of the nine coefficients $k_1-k_9$ may be expressed as a function of the eight other coefficients. The inverse problem is now nonlinear and may be solved by an iterative process. Mathematical detail of this inversion is given in the Appendix with the expression of the first derivative of the quadric equation with respect to the eight remaining coefficients. The error $\sigma_{ncq}$ is calculated following Equation (9) and may be directly compared to the value of $\sigma$ for the fit of the central quadric surface.

## RESULTS

### Digital Shape Models

Samples of shatter cones from nine different impact sites have been studied. Shatter cones from impact structures of different sizes and formed in sedimentary (limestone, dolomite, sandstone, shale) or igneous rocks (basalts, granites) were selected in order to examine whether systematic variations with rock type and crater size may occur. The digital shape models obtained by both techniques are listed in Table 2. In total, six shape models using the Helicon Focus technique, and 14 shape models derived from data generated with the laser-scanning technique are presented. The Helicon Focus shape models of shatter cones from the Gosses Bluff impact structure are derived from images acquired in the field during a visit of D.B. and L.B. to central Australia. The image acquisition procedure was done in the laboratory for the other Helicon Focus shape models. The precision of the shape models is estimated from

comparing the Helicon Focus outputs on the reference sphere with its expected topography (Fig. 2). The estimated value is only valid for the z-domain of half of the reference sphere (0–3 cm), and must be extrapolated elsewhere, as shatter cones are larger than the reference sphere. The precision outside this domain is therefore unknown, and it was assumed to be of the same order of magnitude, but is likely larger than the value estimated from the reference sphere.

Several issues with this technique limited the quality of the geometric information and 20 shape models were removed from analysis because precision was larger than 1 cm. Issues limiting the precision include (1) insufficient number of images acquired in the field, (2) variation of illumination conditions (clouds) during the acquisition procedure, and (3) poor adjustment of the Helicon Focus output to the expected topography of the reference sphere. The precision of selected shape models in the direction of the line of sight ranges from about 5 to 10 mm (Fig. 2). Shape models using Helicon Focus are presented in Fig. 3.

Shape models produced by the laser-scanning technique are presented in Fig. 4. The topographic information is represented using a color-coded shaded relief. Grids were interpolated at 50 µm per pixels and the vertical precision, as given by the manufacturer, is about 25 µm. The resolution and precision of the shape models enable visual inspection of the morphology of the divergent striations. This technique and these data may be used for further studies focusing on the geometric characteristics of striations.

### Quadric Surface Classification

Surface classification results for Helicon Focus and laser-scan shape models are given in Tables 3 and 4, respectively. Five Helicon Focus shape models of six, and 10 of 14 shape models from laser scanning are classified as single-sheet hyperboloid surfaces. The
<figure><img src="image_3.png" /><figcaption>Fig. 2. Calibration curves for the shape models obtained from Helicon Focus. The calibration is obtained from the comparison of $z_{app}$ (as given by Helicon Focus) with the expected elevation on the reference sphere. Dashed black lines correspond to the value of average values of $z_{app}$ in 10 different bins equally spaced over the range of $z_{app}$ values. The red lines indicate the linear or polynomial fit adjusted on the curve of the average values of $z_{app}$.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig. 3. Shape models obtained from Helicon Focus. The black and white version of the sharp image generated for each sample by Helicon Focus has been superposed to a color-coded representation of the digital elevation model. Examples shown relate to shatter cones from (a) Jebal Waqf as Suwwan (36 images), (b) Haughton Dome (45 images), (c) Gosses Bluff (32 images), (d) Gosses Bluff (70 images), (e) Gosses Bluff (40 images), and (f) Gosses Bluff (57 images). The numbers in brackets correspond to the number of images acquired with variable depth-of-focus (DOF) that were processed together by Helicon Focus. The dashed white lines delineate the areas considered for surface classification.</figcaption></figure>

second-most frequent surface type is represented by the
double-sheet hyperboloid, and one Helicon Focus shape
model was classified as an ellipsoid. Examination of

differences between the laser shape models and the
quadric surface indicate that the value of σ arises
essentially from the large-scale deviations from the
<figure><img src="image_5.png" /><figcaption>Fig. 4. Shape models obtained from the laser-scanning technique. Topography is represented using a shaded-relief image superimposed on color-coded elevations. The dashed lines indicate the portions of the Digital Shape Models that were used for surface classification.</figcaption></figure>

quadric surface, especially at the edges of the shape model. However, an important contribution of (much larger) measurement errors is also possible in the case of shape models derived with Helicon Focus. With only one exception at Gosses Bluff, we could therefore conclude that shatter cones are actually hyperboloid surfaces. However, the values of $\sigma_{ncq}$ (for forced

inversion considering noncentral quadric surfaces exclusively) and $\sigma$ (free inversion considering all possible types of quadric surface) are comparable; the residuals of the fit in the case of the noncentral quadric forced inversion are slightly larger than in the case of free inversion. Noncentral quadric surfaces are classified as elliptical and hyperbolic paraboloids. Shape models
<figure><img src="image_6.png" /><figcaption>Fig. 4. Continued</figcaption></figure>
<table><caption>Table 3. Classification of shape models for models obtained using Helicon Focus. Free = linear inversion of the quadric surface equation: all possible quadric surfaces are considered. NCQ = forced noncentral quadratic inversion: the determinant *D* is assumed to be equal to zero and the eight remaining free parameters are determined using a nonlinear iterative least square inversion.</caption><thead><tr><th>Sample name</th><th>Inversion type</th><th>D</th><th>Δ</th><th>Quadric type</th><th>σ or σ<sub>ncq</sub></th><th>a</th><th>b</th><th>c</th></tr></thead><tbody><tr><td rowspan="2">Jebal Waqf as Suwwan WASB62 (laboratory)</td><td>Free</td><td>-1.65 × 10<sup>5</sup></td><td>+3.60 × 10<sup>3</sup></td><td>Single-sheet hyperboloid</td><td>1.72 × 10<sup>-2</sup></td><td>104.6</td><td>19.8</td><td>3.8</td></tr><tr><td>NCQ</td><td>0</td><td>+1.20 × 10<sup>3</sup></td><td>Hyperbolic paraboloid</td><td>1.95 × 10<sup>-2</sup></td><td>75.5</td><td>14.2</td><td>—</td></tr><tr><td rowspan="2">Haughton HounftB52 (laboratory)</td><td>Free</td><td>+1.42 × 10<sup>5</sup></td><td>-1.01 × 10<sup>2</sup></td><td>Single-sheet hyperboloid</td><td>2.17 × 10<sup>-3</sup></td><td>10.7</td><td>1.2</td><td>3.7</td></tr><tr><td>NCQ</td><td>0</td><td>-6.80 × 10<sup>-1</sup></td><td>Elliptical paraboloids (cylinder/plane)</td><td>2.22 × 10<sup>-3</sup></td><td>111.7</td><td>8.7</td><td>—</td></tr><tr><td rowspan="2">Gosses Bluff SC07B52 (field)</td><td>Free</td><td>-1.73 × 10<sup>2</sup></td><td>-4.86</td><td>Ellipsoid</td><td>1.35 × 10<sup>-2</sup></td><td>241.2</td><td>115.0</td><td>12.8</td></tr><tr><td>NCQ</td><td>0</td><td>-3.06 × 10<sup>-2</sup></td><td>Elliptical paraboloid (cylinder/plane)</td><td>1.36 × 10<sup>-2</sup></td><td>127.8</td><td>11.3</td><td>—</td></tr><tr><td rowspan="2">Gosses Bluff SC08B102 (field)</td><td>Free</td><td>+0.52</td><td>2.66 × 10<sup>-4</sup></td><td>Single-sheet hyperboloid (cylinder/plane)</td><td>1.17 × 10<sup>-3</sup></td><td>104.0</td><td>4.0</td><td>38.0</td></tr><tr><td>NCQ</td><td>0</td><td>2.43 × 10<sup>-5</sup></td><td>Elliptical paraboloid (cylinder/plane)</td><td>1.17 × 10<sup>-3</sup></td><td>68.5</td><td>9.0</td><td>—</td></tr><tr><td rowspan="2">Gosses Bluff SC16B32 (field)</td><td>Free</td><td>-1.90</td><td>5.67 × 10<sup>-4</sup></td><td>Single-sheet hyperboloid (cylinder/plane)</td><td>4.46 × 10<sup>-4</sup></td><td>91.5</td><td>19.9</td><td>2.0</td></tr><tr><td>NCQ</td><td>0</td><td>8.21 × 10<sup>-5</sup></td><td>Hyperbolic paraboloid (cylinder/plane)</td><td>9.93 × 10<sup>-4</sup></td><td>312.7</td><td>10.0</td><td>—</td></tr><tr><td rowspan="2">Gosses Bluff SC17B52 (field)</td><td>Free</td><td>-1.95 × 10<sup>-1</sup></td><td>2.69 × 10<sup>-4</sup></td><td>Single-sheet hyperboloid (cone)</td><td>3.83 × 10<sup>-3</sup></td><td>171.2</td><td>47.5</td><td>14.3</td></tr><tr><td>NCQ</td><td>0</td><td>+6.05 × 10<sup>-5</sup></td><td>Hyperbolic paraboloid (cylinder/plane)</td><td>3.82 × 10<sup>-3</sup></td><td>124.3</td><td>34.1</td><td>—</td></tr></tbody></table>

from the laser-scanning technique are classified with one exception as hyperbolic paraboloids (in the case of forced inversion of a noncentral quadric). Figure 5 illustrates the difficulties of classifying quadric surfaces, especially when only portions of the volumes shown in the illustration are represented in the sample, or if topographic measurements have large errors, as in the case of Helicon Focus shape models.

The values of Δ may be used to discuss whether hyperboloid surfaces may be alternatively classified as conical surfaces (or planar/cylindrical). Considering the standard equations for hyperboloid and conical surfaces, the value of Δ is equal to $(1/a^2)(1/b^2)(1/c^2)$. Therefore, even if Δ is non-null, the conical surface might be a good fit if the value of $(1/Δ)^{1/6}$ is much larger than the dimensions of the sample. For most of the free inversions, the values of Δ range from $10^3$ to $10^7$, which implies that $(1/Δ)^{1/6}$ remains below 1 m. It is therefore possible to conclude that conical surfaces do

not correctly represent the shape of these objects, with the exception of shape models from images acquired in the field that could be classified either as planar or cylindrical surfaces. This difference between shape models acquired in the field with those from selected, well-developed shatter cone samples illustrates the fact that near-planar striated surfaces are common in the field—whereas specimens selected for laboratory studies are generally characterized by a more pronounced curvature—a simple bias in the selection of shatter cones.

We have also examined in detail the values of the parameters *a* and *b* of the quadric surfaces in the standard form (Fig. 6a). Objects with similar values for *a* and *b* have a circular base and would likely be qualified as a “complete cone” or “near-complete cone.” When *a* is larger than *b*, the horizontal base (parallel to the *x*,*y* plane) is elliptical. If *a* ≫ *b*, the surface may be described as “near planar,” and retains visible curvature
<table><caption>Table 4. Classification of shape models obtained by the laser-scanning technique for specimens from different impact structures. Free = linear inversion of the quadric surface equation: all possible quadric surfaces are considered. NCQ = forced noncentral quadratic inversion: the determinant *D* is assumed to be equal to zero and the eight remaining free parameters are determined using a nonlinear iterative least square inversion.</caption><thead><tr><th>Sample name</th><th>Inv. type</th><th>D</th><th>Δ</th><th>Quadric type</th><th>σ or σ<sub>ncq</sub></th><th>a (mm)</th><th>b (mm)</th><th>c (mm)</th></tr></thead><tbody><tr><td rowspan="2">Haughton</td><td>Free</td><td>-8.34 × 10<sup>6</sup></td><td>+6.83 × 10<sup>4</sup></td><td>Single-sheet hyperboloid</td><td>1.14 × 10<sup>-2</sup></td><td>7.22</td><td>6.26</td><td>5.69</td></tr><tr><td>NCQ</td><td>0</td><td>+1.88 × 10<sup>4</sup></td><td>Hyperbolic paraboloid</td><td>2.39 × 10<sup>-2</sup></td><td>96.2</td><td>53.5</td><td></td></tr><tr><td rowspan="2">Jebel Waqf<br>as Suwwan</td><td>Free</td><td>-7.46 × 10<sup>7</sup></td><td>+2.14 × 10<sup>6</sup></td><td>Single-sheet hyperboloid</td><td>2.04 × 10<sup>-3</sup></td><td>12.5</td><td>8.22</td><td>5.45</td></tr><tr><td>NCQ</td><td>0</td><td>+2.38 × 10<sup>6</sup></td><td>Hyperbolic paraboloid</td><td>3.60 × 10<sup>-3</sup></td><td>107.3</td><td>52.5</td><td></td></tr><tr><td rowspan="2">Serra da<br>Cangalha</td><td>Free</td><td>+4.19 × 10<sup>6</sup></td><td>+8.93 × 10<sup>3</sup></td><td>Single-sheet hyperboloid</td><td>1.37 × 10<sup>-3</sup></td><td>24.6</td><td>0.4</td><td>4.9</td></tr><tr><td>NCQ</td><td>0</td><td>+2.71 × 10<sup>3</sup></td><td>Hyperbolic paraboloid</td><td>1.90 × 10<sup>-3</sup></td><td>28.3</td><td>2.5</td><td></td></tr><tr><td rowspan="2">Rochechouart 1</td><td>Free</td><td>-4.31 × 10<sup>6</sup></td><td>-1.22 × 10<sup>4</sup></td><td>Double-sheet hyperboloid</td><td>7.44 × 10<sup>-2</sup></td><td>20.1</td><td>8.9</td><td>0.4</td></tr><tr><td>NCQ</td><td>0</td><td>+9.13 × 10<sup>4</sup></td><td>Hyperbolic paraboloid</td><td>7.45 × 10<sup>-2</sup></td><td>150.4</td><td>6.7</td><td></td></tr><tr><td rowspan="2">Rochechouart 2</td><td>Free</td><td>+1.60 × 10<sup>6</sup></td><td>-1.16 × 10<sup>4</sup></td><td>Double-sheet hyperboloid</td><td>5.12 × 10<sup>-2</sup></td><td>42.8</td><td>2.0</td><td>5.6</td></tr><tr><td>NCQ</td><td>0</td><td>+1.53 × 10<sup>4</sup></td><td>Hyperbolic paraboloid</td><td>5.14 × 10<sup>-2</sup></td><td>41.0</td><td>14.8</td><td></td></tr><tr><td rowspan="2">Steinheim<br>Basin 1</td><td>Free</td><td>-2.68 × 10<sup>5</sup></td><td>+1.49 × 10<sup>2</sup></td><td>Single-sheet hyperboloid</td><td>2.48 × 10<sup>-2</sup></td><td>30.7</td><td>3.8</td><td>0.2</td></tr><tr><td>NCQ</td><td>0</td><td>+2.12 × 10<sup>2</sup></td><td>Hyperbolic paraboloid</td><td>2.48 × 10<sup>-2</sup></td><td>35.3</td><td>1.9</td><td></td></tr><tr><td rowspan="2">Steinheim<br>Basin 3</td><td>Free</td><td>+8.61 × 10<sup>4</sup></td><td>+3.13 × 10<sup>2</sup></td><td>Single-sheet hyperboloid</td><td>3.98 × 10<sup>-3</sup></td><td>64.0</td><td>1.1</td><td>10.9</td></tr><tr><td>NCQ</td><td>0</td><td>+3.80 × 10<sup>2</sup></td><td>Hyperbolic paraboloid</td><td>4.10 × 10<sup>-2</sup></td><td>43.6</td><td>5.6</td><td></td></tr><tr><td rowspan="2">Steinheim<br>Basin 4</td><td>Free</td><td>-3.47 × 10<sup>5</sup></td><td>+5.40 × 10<sup>3</sup></td><td>Single-sheet hyperboloid</td><td>2.17 × 10<sup>-2</sup></td><td>42.8</td><td>10.9</td><td>7.0</td></tr><tr><td>NCQ</td><td>0</td><td>+1.21 × 10<sup>1</sup></td><td>Hyperbolic paraboloid</td><td>2.24 × 10<sup>-2</sup></td><td>18.2</td><td>10.2</td><td></td></tr><tr><td rowspan="2">Vargeão<br>Dome</td><td>Free</td><td>-1.33 × 10<sup>6</sup></td><td>+3.15 × 10<sup>4</sup></td><td>Single-sheet hyperboloid</td><td>3.91 × 10<sup>-2</sup></td><td>42.8</td><td>25.1</td><td>2.9</td></tr><tr><td>NCQ</td><td>0</td><td>+1.09 × 10<sup>3</sup></td><td>Hyperbolic paraboloid</td><td>4.00 × 10<sup>-2</sup></td><td>76.3</td><td>8.5</td><td></td></tr><tr><td rowspan="2">Vista Alegre</td><td>Free</td><td>-2.41 × 10<sup>7</sup></td><td>-4.25 × 10<sup>5</sup></td><td>Double-sheet hyperboloid</td><td>3.62 × 10<sup>-2</sup></td><td>31.8</td><td>12.2</td><td>1.2</td></tr><tr><td>NCQ</td><td>0</td><td>+4.68 × 10<sup>3</sup></td><td>Hyperbolic paraboloid</td><td>3.64 × 10<sup>-2</sup></td><td>88.3</td><td>4.4</td><td></td></tr><tr><td rowspan="2">Vredefort 1</td><td>Free</td><td>-5.50 × 10<sup>8</sup></td><td>+2.12 × 10<sup>7</sup></td><td>Single-sheet hyperboloid</td><td>5.23 × 10<sup>-2</sup></td><td>18.0</td><td>6.4</td><td>2.8</td></tr><tr><td>NCQ</td><td>0</td><td>+4.69 × 10<sup>7</sup></td><td>Hyperbolic paraboloid</td><td>5.56 × 10<sup>-3</sup></td><td>81.7</td><td>33.2</td><td></td></tr><tr><td rowspan="2">Vredefort 3</td><td>Free</td><td>+3.55 × 10<sup>8</sup></td><td>-3.79 × 10<sup>7</sup></td><td>Double-sheet hyperboloid</td><td>5.91 × 10<sup>-2</sup></td><td>12.7</td><td>0.5</td><td>8.6</td></tr><tr><td>NCQ</td><td>0</td><td>+5.64 × 10<sup>5</sup></td><td>Hyperbolic paraboloid</td><td>5.96 × 10<sup>-2</sup></td><td>66.7</td><td>4.1</td><td></td></tr><tr><td rowspan="2">Vredefort 4</td><td>Free</td><td>-8.39 × 10<sup>6</sup></td><td>+1.53 × 10<sup>5</sup></td><td>Single-sheet hyperboloid</td><td>3.24 × 10<sup>-2</sup></td><td>32.4</td><td>9.2</td><td>2.7</td></tr><tr><td>NCQ</td><td>0</td><td>+5.17 × 10<sup>4</sup></td><td>Hyperbolic paraboloid</td><td>3.25 × 10<sup>-2</sup></td><td>59.5</td><td>16.0</td><td></td></tr><tr><td rowspan="2">Vredefort 5</td><td>Free</td><td>-6.50 × 10<sup>8</sup></td><td>+1.66 × 10<sup>7</sup></td><td>Single-sheet hyperboloid</td><td>3.73 × 10<sup>-2</sup></td><td>18.0</td><td>4.7</td><td>1.9</td></tr><tr><td>NCQ</td><td>0</td><td>-2.85 × 10<sup>4</sup></td><td>Elliptical paraboloid</td><td>7.74 × 10<sup>-2</sup></td><td>202.5</td><td>6.3</td><td></td></tr></tbody></table>

only along *z*-profiles. The values of the two parameters range from about 10 mm to more than 100 mm, as expected from visual inspection of the curvature of the selected samples. Several samples have *a/b* ratios close to 1 (e.g., the shatter cones from Haughton or Jebel Waqf as Suwwan), but others have *a/b* ratios greater than 10. Large *a* and *b* values, and high *a/b* ratios are common for shape models of shatter cones examined in the field at Gosses Bluff.

There are no clusters or notable trends with respect to the various types of host lithologies. This appears consistent with the impression in the field that poorly developed shatter cones (or near-planar surfaces) often coexist with curved surfaces. The *c* values are related to the apparent angle of the apex for conical or hyperboloid surfaces. The values of *c* versus *b* are represented in Fig. 6b. The values of *c* are spread over two orders of magnitude, reflecting the wide range of apical angles (60°–120°) that have been already reported by other studies (e.g., Baratoux and Reimold [2016] and

references therein; Sagy et al. 2004). Here again, it is not possible to identify any trend with respect to lithology. Crater size does not appear to control the parameters of the quadric surface either (Fig. 6b). For each crater, a wide range of values for *b* is obtained. This absence of correlation between quadric parameters and crater size is consistent (Fig. 7) with the fact that no qualitative correlation between the apparent dimensions and geometric characteristics of shatter cones and crater sizes has been previously reported.

In summary, we conclude that shatter cones may well be described by either paraboloid or hyperboloid surfaces, although the classification procedure indicates that single-sheet hyperboloids produce the best matches with our shape models, and especially in the cases of the most precise ones acquired with the laser-scanning technique. There is no obvious trend between the characteristics and parameters of quadric surfaces with rock type or crater size. However, the number of shape models is probably too small for investigating any
<figure><img src="image_7.png" /><figcaption>Fig. 5. 3-D color view of the most frequent quadric surfaces found after application of the classification procedure applied to the shape models of shatter cones. The blue-red color scale indicates the value (elevation) in the vertical direction (z-axis). The figures illustrate the difficulties of discriminating between these different types of surfaces when only portions of the represented volumes are available (as in the case of incomplete cones), in the case of large measurement errors, or in the presence of geometric imperfections that are often observed in natural samples.</figcaption></figure>

systematic lithological control on the geometry of shatter cones. Moreover, other parameters such as distance to the center of the crater, or shock intensity may also control the shape of shatter cones, which complicates the search for lithological-dependent effects. Conical surfaces do not represent a common shape for these objects. This result confirms the observation of curved linear profiles in the 2-D measurements from Sagy et al. (2004).

DISCUSSION

Five different hypotheses (Johnson and Talbot 1964; Gash 1971; Sagy et al. 2002; Baratoux and Melosh 2003; Dawson 2009) have been proposed to explain the formation of shatter cones. They are reviewed in some detail in the introduction paper of this special issue (Baratoux and Reimold 2016). The

hypotheses of Johnson and Talbot (1964) and Gash (1971) have been discarded based on shock physics (Baratoux and Melosh 2003; Baratoux and Reimold 2016), so that the discussion is not repeated here. Dawson (2009) proposed that shatter cones result from shear banding. His numerical simulations show how the deformation is localized when a shock wave interacts with material heterogeneities. However, the zone of intense deformation is not necessarily associated with a fracture, as adiabatic shear bands usually retain material continuity. Furthermore, deformation is localized along a conical surface, whereas shape models produced and examined in this study are generally not consistent with cones. We will, therefore, examine below in more detail whether and how the new three-dimensional data may be used to test the hypotheses of Baratoux and Melosh (2003) and Sagy et al. (2002).
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. a) Representation of *a* versus *b* parameters of the fitted quadric surfaces for all analyzed shatter cone samples. Symbols correspond to the quadric surface types, and colors correspond to the rock type. A character “H” has been added as symbol for shape models derived using Helicon Focus, other symbols correspond to shape models derived with the laser-scanning technique. b) Representation of *b* versus *c* for free inversion. Colors and symbols as in (a).</figcaption></figure>

Baratoux and Melosh (2003) proposed that shatter cone surfaces develop as the result of an interference process between the main shock wave and scattered waves emerging from small-scale variations of the elastic properties of target rock. They have simulated how damage is localized when a shock wave interacts with scattered elastic waves that develop around a material heterogeneity (such as an inclusion of a distinct material in a homogeneous rock). The surface of the fracture corresponds to the locus of a constructive interference between the extensive hoop stress in the scattered wave and the tail of the main shock wave where the hoop stress becomes also extensive (Baratoux and Melosh 2003; see Baratoux and Reimold [2016] for details). The zones of intense damage presented in the axisymmetric simulation of Baratoux and Melosh (2003)

<figure><img src="image_10.png" /><figcaption>Fig. 7. Representation of *b* versus impact structure diameter. Colors and symbols as in Fig. 6a.</figcaption></figure>

are slightly curved. However, it is difficult to draw a definitive conclusion from visual inspection of the figures representing damage as a function of time; the resolutions of the simulations are too coarse for this purpose. Alternatively, the shape of the surface where damage has accumulated until rupture may be predicted from a simple analytical model (Fig. 8). At great distance from the center of the crater, and in the vicinity of a small inclusion, the spherical shock wave may be approximated by a planar front wave propagating in the *z* direction. The position of the front of the shock wave as a function of time is given by:

$$z = vt + z_0 \quad (10)$$

where *v* is the velocity of the shock wave and *t* the time since the front wave was at the position *z* = *z*₀. For simplicity, we take an arbitrary reference *z*₀ = 0 when *t* = 0 at the center of the material heterogeneity. The position of the extensive scattered wave corresponding to the position of the peak of extensive hoop stress is given by:

$$x^2 + y^2 + z^2 = [v(t + \tau)]^2 \quad (11)$$

where *τ* is the time delay between the shock front and the tail of the shock wave where the hoop stress becomes extensive enough, so that the extensive stress resulting from the constructive interference exceeds the strength of the material in tension (Baratoux and Melosh 2003). Combining Equation 10 with Equation 11, we derive the equation of the intersection points between the main shock wave and the scattered wave where constructive interference occurs and results in localized accumulation of damage:
<figure><img src="image_11.png" /><figcaption>Fig. 8. Illustration of the analytical model used to predict the shape of shatter cones from the interference model of Baratoux and Melosh (2003). The material inclusion is represented by a gray circle at the center of the sketch. The black circles represent the position of the scattered wave at the times $t_1$ and $t_2$. Light gray planes represent the position of the tail of the main shock wave, when the hoop stress becomes extensive enough, so that the combination of stresses resulting from the main and scattered waves exceeds the resistance of the material in tension. The loci of the circular intersections in cross section are represented by dotted lines. The thick black line represents the predicted shape of the fracture in the $y-z$ plane, resulting from the localization of damage associated with occurrence of extensive stresses beyond the resistance of the material in tension.</figcaption></figure>

$$ \frac{x^2}{\tau^2 v^2} + \frac{y^2}{\tau^2 v^2} - \frac{2z}{\tau v} = 1 \quad (12) $$

It is of note that Equation 12 is the standard form of the equation of a circular paraboloid, where $a = b = (\tau v/2)$. The predicted shape is, therefore, different from the most common surfaces found in this study. However, the analytical model makes several important approximations. The extensive hoop stress in the scattered wave decreases with distance from the heterogeneity (which is neglected in Equation 12), and a replace by larger value of the extensive hoop stress may be required to overcome the strength of the material in tension as the fracture develops. The analytical model does not consider the possible resulting increase in $\tau$ with time. In addition, the analytical model does not take into account the fact that the stress field is modified when

damage occurs, and new numerical simulations would be needed to evaluate the consequence of this approximation on the analytically predicted shape of fractures (Melosh, personal communication). Last but not least, the analytical model applies in the context of a punctual material inclusion. Different shapes may be obtained for linear or more irregular, or even planar or curviplanar (fractures, joints) heterogeneities, opening the possibility to produce a wide range of geometric properties with this model. 3-D numerical simulations at higher resolution than those presented in Baratoux and Melosh (2003) and considering different geometric characteristics of material inclusions would be needed to further clarify the influence of the shapes and natures of heterogeneities on the geometry of striated surfaces.

Sagy et al. (2002) proposed that shatter cones form a network of rapidly propagating branching fractures, whereas striations would result from front waves generated when the fracture front reaches material heterogeneities. Experimental studies show that rapidly branching fractures associated with shock waves are curved and would, therefore, be qualitatively compatible with the shape models obtained in this study. Branching fractures were described as “spoon-like” by Sagy et al. (2004), reflecting the fact that their shape cannot be predicted based on theoretical grounds. Empirical power-law relationships were derived using 2-D topographic profiles (Sharon and Fineberg 1996). A theoretical background to account for the shape of the branching fracture has yet to be developed. 3-D observations of the shape of these branching fractures are not available. The new data presented in this paper must await the development of a theory explaining the 3-D shape of the branching fractures. Alternatively, experiments with acquisition of appropriate geometric data would allow direct comparisons with the 3-D observations on shatter cones and a quantitative assessment of the hypothesis that shatter cones are branching fractures.

## CONCLUSIONS

We have produced here the first series of digital shape models of natural shatter cones associated with impact structure of different sizes (3.8–160 km diameters) and formed in different rock types (basalts, granite, dolomite, limestone, shale, and sandstone). Two methods of shape reconstruction have been tested. The method based on the laser-scanning technique produces models of much higher precision than the one based on imagery, but is not easily applied in the field. Near-planar surfaces are reported from field observations. These high-resolution measurements are, therefore, useful to discuss the shape of shatter cones and whether these shapes may
be represented by a unique family of surfaces. We have then proposed that these shape models may be approximated by one or several types of quadric surfaces and we have classified them according to the best match with a given quadric surface type. It is found that shatter cones are well described by quadric surfaces and are generally classified as either paraboloid or hyperboloid surfaces. The most frequent classification appears to be the single-sheet hyperboloid. Therefore, the shape of these striated surfaces occurring exclusively in the impact context, and usually described as shatter cones, are actually not conical (considering the mathematical definition of a conical surface). No particular correlation has been found between quadric types or parameters with impact crater size or lithology. Further examination of the shape models will be possible in the future, such as quantifications of curvatures, or detailed analyses of striations that are well described by the shape models produced with the laser-scanning technique.

Unfortunately, it is not possible yet to derive a definitive conclusion with respect to existing hypotheses of shatter cone formation, based on these new data. Further theoretical or experimental work and numerical simulations are needed. New numerical simulations at higher resolution (and in 3-D) are needed to determine the shape of the surface of the fracture according to the hypothesis of Baratoux and Melosh (2003). In the absence of theoretical grounds to predict the 3-D shape of branching fractures, new experiments and appropriate measurements of the shape of the surface of branching fractures are needed for direct comparison with the macroscopic geometric properties of shatter cones.

*Acknowledgments*—The acquisition of images for the reconstruction of shape models of shatter cones at Gosses Bluff were achieved during a visit of D.B. and L.B. to the University of Western Australia, Centre for Exploration Targeting. This research was funded by the National Program of Planetary Science of INSU (France). WUR's research is supported by the Deutsche Forschungsgemeinschaft (DFG) and the Museum für Naturkunde Berlin. The constructive comments of Stephan Hergarten and an anonymous reviewer improved the presentation of this research. The shape

# APPENDIX

## Determination of the Parameters for a Noncentral Quadric Surface

Noncentral quadric surfaces have the determinant D (Equation 3) equal to 0 and $k_1$ may be expressed as a function of $k_2 - k_9$:

models are available on the Zenodo data repository (https://zenodo.org/record/35029).

*Editorial Handling*—Dr. Natalia Artemieva

## REFERENCES

Baratoux D. and Melosh H. J. 2003. The formation of shatter cones by shock wave interference during impacting. *Earth and Planetary Science Letters* 216:43–54.

Baratoux D. and Reimold W. U. 2016. Introduction to the special issue. Shatter cones: An overview of occurrences and genetic hypotheses. *Meteoritics & Planetary Science*.

Branco W. and Fraas E. 1905. *Das kryptovulkanische Becken von Steinheim*. Berlin: Abhandlungen der Königl. Preuss. Akademie der Wissenschaften, Verlag G. 64 pp.

Dai M., Newman T. S., and Cao S. 2007. Least-squares-based fitting of paraboloids. *Pattern Recognition* 40:504–515.

Dawson E. 2009. Meteorite impact shatter cones—Adiabatic shear bands? 9th International Conference on the Mechanical and Physical Behavior of Materials under Dynamic Loading, Brussels, Belgium.

Ferrière L. and Osinski G. 2012. Shock metamorphism. In *Impact cratering: Processes and products*, edited by Osinski G. R. and Pierazzo E. Hoboken, New Jersey: John Wiley & Sons, Ltd. doi:10.1002/9781118447307.ch8

Gash P. J. S. 1971. Dynamic mechanism for the formation of shatter cones. *Nature Physical Science* 230:32–35.

Hall E. L., Tio J. B. K., McPherson C. A., and Sadjadi F. A. 1982. Measuring curved surfaces for robot vision. *IEEE Computer* 15:42–54.

Johnson G. and Talbot R. 1964. A theoretical study of the shock wave origin of shatter cones. M.S. Thesis, Air Force Institute of Technology, Wright-Patterson Air Force Base, Dayton, Ohio. 170 pp.

Sagy A., Reches Z., and Fineberg J. 2002. Dynamic fracture by large extraterrestrial impacts as the origin of shatter cones. *Nature* 418:310–313.

Sagy A., Fineberg J., and Reches Z. 2004. Shatter cones: Branched, rapid fractures formed by shock impact. *Journal of Geophysical Research (Solid Earth)* 1209:10209.

Sharon E. and Fineberg J. 1996. Microbranching instability and the dynamic fracture of brittle materials. *Physical Review B (Condensed Matter)* 54:7128–7139.

Wilk J. and Kenkmann T. 2015a. The surface structure of shatter cones in experimental impact craters (abstract #2637). 46th Lunar and Planetary Science Conference. CD-ROM.

Wilk J. and Kenkmann T. 2016. Formation of shatter cones in MEMIN Impact experiments. *Meteoritics & Planetary Science*. doi: 10.1111/maps.12682

$$k_1 = \frac{k_2 k_6^2 + k_3 k_4^2 - k_4 k_5 k_6}{4 k_2 k_3 - k_5^2} \quad (A1)$$

which may be calculated as long as $4k_2k_3 - (k_5)^2$ is not zero.

In the nonlinear least-squares method, the parameters $k_2, \dots, k_9$ expressed as the vector $\mathbf{k}$ are
determined by an iterative approach, and the correction **dk**<sup>j</sup> calculated at the iteration j is added to determine a new set of parameters **k**<sup>j+1</sup>.

$$
\mathbf{k}^{j+1} = \mathbf{k}^j + \mathbf{dk}^j \quad (A2)
$$

where

$$
\mathbf{dk}^j = \begin{pmatrix} dk_2^j \\ dk_3^j \\ dk_4^j \\ dk_5^j \\ dk_6^j \\ dk_7^j \\ dk_8^j \\ dk_9^j \end{pmatrix} \tag{A3}
$$

Equation 1 may be rewritten for each point $(x_i, y_i, z_i)$ and for a given set of $k_2, \dots, k_9$ parameters at the iteration $j$:

$$
\begin{equation}
\begin{aligned}
f_i^j = k_2^j y_i^2 &+ k_2^j z_i^2 + k_2^j x_i y_i + k_2^j y_i z_i + k_2^j x_i z_i + k_2^j x_i \\
& + k_2^j y_i + k_2^j z_i + \frac{u^j}{v^j} x_i^2 = 1
\end{aligned}
\tag{A4}
\end{equation}
$$

where

$$
\frac{u^j}{v^j} = \frac{k_2^j (k_6^j)^2 + k_3^j (k_4^j)^2 - k_4^j k_5^j k_6^j}{4 k_2^j k_3^j - (k_5^j)^2} = k_1^j \quad (A5)
$$

Equation A4 may be linearized and residuals are
given by:

$$
\mathbf{v}^j = \mathbf{A}^j \mathbf{d} \mathbf{k}^j - \mathbf{y}^j \quad (A6)
$$

where

$$
\mathbf{y}_j = \begin{pmatrix} 1 - f_1^j \\ \vdots \\ 1 - f_i^j \\ \vdots \\ 1 - f_n^j \end{pmatrix} \quad (A7)
$$

and

$$
A^j = \begin{pmatrix}
\frac{\partial f_1^j}{\partial k_2} & \frac{\partial f_1^j}{\partial k_3} & \frac{\partial f_1^j}{\partial k_4} & \frac{\partial f_1^j}{\partial k_5} & \frac{\partial f_1^j}{\partial k_6} & \frac{\partial f_1^j}{\partial k_7} & \frac{\partial f_1^j}{\partial k_8} & \frac{\partial f_1^j}{\partial k_9} \\
\frac{\partial f_2^j}{\partial k_2} & \frac{\partial f_2^j}{\partial k_3} & \frac{\partial f_2^j}{\partial k_4} & \frac{\partial f_2^j}{\partial k_5} & \frac{\partial f_2^j}{\partial k_6} & \frac{\partial f_2^j}{\partial k_7} & \frac{\partial f_2^j}{\partial k_8} & \frac{\partial f_2^j}{\partial k_9} \\
\frac{\partial f_3^j}{\partial k_2} & \frac{\partial f_3^j}{\partial k_3} & \frac{\partial f_3^j}{\partial k_4} & \frac{\partial f_3^j}{\partial k_5} & \frac{\partial f_3^j}{\partial k_6} & \frac{\partial f_3^j}{\partial k_7} & \frac{\partial f_3^j}{\partial k_8} & \frac{\partial f_3^j}{\partial k_9} \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
\frac{\partial f_n^j}{\partial k_2} & \frac{\partial f_n^j}{\partial k_3} & \frac{\partial f_n^j}{\partial k_4} & \frac{\partial f_n^j}{\partial k_5} & \frac{\partial f_n^j}{\partial k_6} & \frac{\partial f_n^j}{\partial k_7} & \frac{\partial f_n^j}{\partial k_8} & \frac{\partial f_n^j}{\partial k_9}
\end{pmatrix} \quad (A8)
$$

The corrections to be applied at each iteration *j* are
given by:

$$
\mathbf{dk}^j = (\mathbf{A}^j \mathbf{T} \mathbf{A}^j)^{-1} \mathbf{A}^j \mathbf{T} \mathbf{y}_j \quad (A9)
$$

The expression of the partial derivatives in the
matrix *A* are given below (the exponent *j* referring to a
given iteration has been omitted).

$$
\frac{\partial f_i}{\partial k_2} = y_i^2 + \frac{k_6^2 v - 4 k_3 u}{v^2} x_i^2 \quad (A10)
$$

$$
\frac{\partial f_i}{\partial k_3} = z_i^2 + \frac{k_4^2 v - 4 k_2 u}{v^2} x_i^2 \quad (A11)
$$

$$
\frac{\partial f_i}{\partial k_4} = x_i y_i + \frac{2k_3 k_4 - k_5 k_6}{v} x_i^2 \quad (A12)
$$

$$
\frac{\partial f_i}{\partial k_5} = y_i z_i + \frac{k_4 k_6 v - 2 k_5 u}{v^2} x_{ii}^2 \quad (A13)
$$

$$
\frac{\partial f_i}{\partial k_6} = x_i z_i + \frac{2k_2 k_6 - k_4 k_5}{v} x_i^2 \quad (A14)
$$

$$
\frac{\partial f_i}{\partial k_7} = x_i \qquad (A15)
$$

$$
\frac{\partial f_i}{\partial k_8} = y_i \qquad (A16)
$$

$$
\frac{\partial f_i}{\partial k_9} = z_i \qquad (A17)
$$