Computer-Based Analysis

The previously described mapping information is the basis for determining the radii of curvature of the curved ridges, and subsequently the mass-wasting

6.3. Computer-Based Analysis velocities, related by Equation3.26. The following sections explain how the velocities were derived and how basic assumptions on Vesta’s shape and rotation influence the analysis.

6.3.1. Defining the Reference Frame

In order to find the radius of curvature of the curved ridges, the mapping on the reference sphere needed to be transferred to a reference frame describing Vesta’s real shape more adequately. Based on the results of the DTM construction, Vesta is best described by an ellipsoid with axes of 285.4 km × 277.7 km × 223.8 km [Preusker, 2013, personal communication]. However, to simplify matters a biaxial reference ellipsoid with the average of the semi- major axes of 281.55 km and a semi-minor axis of 223.8 km was used instead. This approximation is suitable, as the difference to the ellipsoidal semi-major axes is only about 1%. It also extensively simplifies the analysis because it becomes independent of the longitude.

In order to project each mapping point from the 255 km reference sphere onto the 281.55 km × 223.8 km reference biaxial ellipsoid the geocentric coordinates were simply adopted. In other words, the centres of both bodies coincided with the centre of projection. Because this procedure was previously done in reverse when constructing the mosaic (Section 4.2), the technique resulted in a projection from Vesta’s real shape to Vesta’s ellipsoidal shape. However, the following analysis requires geodetic coordinates for determin- ing the mass-wasting velocities from the radii of curvature of the mapped curved ridges (Equation 3.26), because of the relation between geodetic co- ordinates and the surface normal and tangent plane of the spheroid. Thus, Equation 3.15 was applied to the geocentric coordinates of each mapping point to transform them to geodetic coordinates.

6.3.2. Calculating the Velocity

A few terms need to be defined prior the following explanations: As described above, each curved ridge was mapped and approximated by 105 to 757

mapping points depending on their length and mapping accuracy. Considering

an arbitrary mapping point MP, a set of two mapping points with the same point distance from a MP is called the neighbouring point pair of MP (Figure 6.3). For example, the two direct adjacent mapping points of MP are the first order neighbouring point pair. The second order neighbouring point pair includes the points adjacent to the first order neighbouring point pair and so on. Thus, each mapping point can have as many neighbouring point

1st _order neighbouring point pair 5th 4th 3rd 2nd MP

Figure 6.3.: Sketch illustrating the neighbouring point pairs of mapping point MP (blue cross) on a curved ridge (red). The colours and loops join the two members of a neighbouring point pair. For illustrative reasons, only a subset of mapping points (coloured crosses on red line) forming the mapped curved ridge are sketched.

pairs as it is away from the end or beginning of the curved ridge. In other words, the nth point along a curved ridge has either (n − 1) or the (total

amount of points − n) neighbouring point pairs, whichever is smaller. Note

that because the mapping was based on the geological analysis, the actual distance between a mapping point and the members of a neighbouring point pair can vary and is different for each mapping point.

The latitude and local curvature of each mapping point enables the deriva- tion of a mass motion velocity according to Equation 3.26. To analyse the curvature at a mapping point, the tangent plane of the bi-axial ellipsoid at each mapping point’s location was determined. The normal vector of the tangent plane is described by the vector spanned by the longitude and geodetic latitude of the mapping point. All neighbouring point pairs were then projected into this plane. Projecting the neighbouring point pairs into the tangent plane was necessary because the derivation of the inertial circles is in the body-fixed coordinate system of the mapping point with the x- and y-axes spanning the tangent plane (Section 3.2.3). Furthermore, if using the position on the spheroid instead of the tangent plane, the curvature of the

6.3. Computer-Based Analysis

R

(a) (b) (c)

Figure 6.4.: Three steps to determine the unique circle that fits through three points in a plane. (a) Three points in a plane. (b) After connecting two points with one another, two perpendicular bisector are drawn. (c) The two perpendicular bisectors intersect at the centre of the circle that cuts through all three points. The radius is given by the distance from the circle’s centre to any of the three points.

spheroid would limit the circles’ radii.

For each neighbouring point pair and the mapping point, the unique circle with all three points lying on the circumference was determined. Figure 6.4

illustrates the technique. This circle was considered to be the inertial circle related to the Coriolis Effect at the length-scale given by the neighbouring point pair.

The normal vector to the vectors from the mapping point to the previous and the following neighbouring point pair member was determined. If this normal vector pointed outside the ellipsoid the radius at the mapping point was defined as positive. A negative radius was assigned if the normal vector pointed inwards. In this way, curvatures due to masses moving towards the rotation axis were defined as positive and curvatures due to masses moving away from the axis were defined as negative.

The radius of the inertial circle - typically of the order of ∼50 km - and the geodetic latitude of the mapping point were finally used to determine the velocity, by applying Equation 3.26. Consequently, each mapping point was assigned a set of velocities - one velocity derived from each neighbouring point pair.

Note that the distance between the two members of the neighbouring point pairs sets the lower limit for the velocity: the smallest circle which can be described by the neighbouring point pair is the circle with a diameter equal to the distance between the members. For instance, the minimum distance

between the two members of a point pair is ∼300 m. At 45◦S this results in a threshold velocity of ∼0.1 m/s. Velocities below this threshold cannot be resolved by this method. With higher order neighbouring point pairs this threshold grows linearly with the distance between the members of the pair.

The distance between the two members of the neighbouring point pairs also determines the distance over which the velocity is assumed to be constant. A change in velocity is represented by a change in the inertial radius. However, it is uncertain over which distances the velocities producing the curved ridges in the Rheasilvia basin are constant. The masses are accelerated by gravity but also decelerated by friction or topography. Therefore, the most commonly occurring velocity was determined for each mapping point, filtering out infrequent results.

A histogram of all calculated velocities was produced for each mapping point, to determine the most commonly occurring velocity at that mapping point. The bin size was set to 5 m/s, 10% of the simulated velocity of 50 m/s, and the effect of changing the bin size is discussed in Section6.5. All values in the most populated bin were considered to be within the most commonly occurring velocity range. If there was more than one most populated bin - a scenario especially relevant for mapping points with only few neighbouring point pairs (the points near the beginning or end of a curved ridge) - all relevant bins were considered. The average of all values in the most populated bin(s) were stored as the velocity of each mapping point.

Summarizing this procedure. the radii of curvature of each mapping point were determined at various length-scales, using Equation 3.26 these radii were then converted into mass-wasting velocities and subsequently the most commonly occurring velocity over all length-scales was determined by using a histogram and recorded as the “most likely” velocity of the mapping point.

The pseudo IDL-code used to perform this analysis is shown in Ap- pendixA.2.