Multi-fractal-interslipface Angle Curves of a Morphologically Simulated Sand Dune

A sand dune is simulated by means of a non-linear mathematical morphological transformation of which the fractal dimensions with corresponding interslipface angles are computed. This exercise has relevance to test the Validity of the model by 
considering various time series sand dune data that can be retrieved from the robust satellite remote sensing sensors.

In this communication, a supply point of the sand that acts as an initiator that receives the accumu- lated sand due to interferential wind forces forms the symmetrical sand dunes.This process is simu- lated by applying iterative morphological dilation by following a non-linear morphological transfor- mation with respect to a generator which is also represented in morphological terms that posses certain characteristic information like an origin direction and orientation.In order to simulate the symmetrical sand dune in discrete space, an invert- ed triangular type of generator with center as origin and bottom as the vertices is used.At each iteration the fractal dimension has been computed and at the nth generation of the simulation the fractal dimension is found to be converged to the value in two dimensional case.This has been represented in a multifractal curve by a graphical plot between the fractal dimensions at respective iteration of morphological dilation and the degree of multi- scale dilations.This rapid communication contains three brief parts.The first part gives a brief intro- duction to the multiscale morphological dilation [1].In the second part the simulated sand dune with the results are presented both mathematically and diagramatically for a better understanding of the simulated model and concluded with a further scope from the remote sensing perspective.Dilation (5), besides erosion, opening and closing [1], is one of the simplest quantitative 72 B.S. DAYA SAGAR morphological set transformations.The discrete binary image, M, is defined as a finite subset of Euclidean two dimensional space, 1R 2 that can admit values between and 0. Hereafter the ini- tiator refers to as M that is a supply point in the present context.Prior to understand the simu- lation process morphological dilation is defined as a set transformation that expand a set in a mean- ingful way.This morphological transformation can be visualized as working with two images.They are initiator M, and a generator that is represented as S. Each S has a designed shape that can be thought of as a probe of the M.This basic morphological transformation, i.e., dilation besides erosion, and opening and closing can be defined with respect to the generator S with scaling factor e, supply point M. The multiscale morphological dilation is math- ematically represented as follows.
Dilation: (5(M) {m: Sm ( M} Us slVI, where e-1,2,...,n (1) where S (s s c S ), i.e., S rotated 180 around the origin and S transpose of S in this context, then the morphological dilation is dissimilar to Minkowski addition as is the case when S= transpose of S.
This transformation is iteratively performed on the supply point, in other words an initiator, a dot in the two dimensional space to have differ- ent degrees of conical sand dunes with different peaks by means of a generator which is in an inverted triangle in shape.In contrast the initi- ator is the supply area depicted as a base with stationary length in the earlier work [2].Due to the simplicity in description the shape of the gen- erator at different scales is not described diagramatically.Figure gives a consolidated diagram that consists of coded conical sand dune and overlaid systematically (upto 25 iterative dilations of initiator by a generator).This simulated conical sand dune, which is exactly similar to the one simulated by Bertz et al. [3], also consists of the simulated conical sand dune at

FIGURE
The sand dune shown is the digital image simulated from supply point (initiator) and a generator (inverted triangle) with center as origin and bottom as vertices.Simulation of sand dune is shown at different degrees of morphological dilations of initiator by generator.The fractal dimensions at the respective growth levels of Sand dune are shown at the right side of the axis.At the left, the interslipface angles have been depicted for respective degree of sand dune growth.To have a visual appeal, the simulated sand dune is enlarged.Hence the diagram at the high iterates of dilations are protruding, in contrast to the interslipface angles mentioned.respective degree of growth of sand dunes.Further a graphical plot is shown between the dilation degree and the corresponding fractal dimen- sions of sand dunes at the corresponding degree of dilations in Figure 2a.This is done to have a visual appeal to understand what author intends to say.The interslipface angle [2] at correspond- ing level of sand dune in the growth process simulated through multiscale dilations has been computed by following Eq.( 2) proposed by Sagar [2]. a {LogN/Log[2 Sin(0/2)]} Dr (2)   where a+DT--D; N= 2 (2 slipfaces for the sand dune) From Eq. ( 2), it is reduced to compute the interslipface angles (0) by incorporating the   (Fig. 2a, b), between the degree of dilation, and fractal dimensions [4] as well as corresponding interslipface angles.The interslipface angle limits are between 97 and 157 of which the corre- sponding fractal dimensions range between 1.7 and for this morphologically simulated sand dune growth.It is worth mentioned here, as the base length of the sand dune increases, the dune becomes flattened in this simulated sand dune model.These results are sensitive with a variation in any of the characteristic information such as origin, orientation, direction of the generator that plays a vital role in the simulation process.With this initial information and type of sand dune, morphological evolution can be quantified pre- cisely to understand the nature or phenomenon of self organized criticality.With the advent of re- cent advances in the generation of high resolution digital elevation models from the remotely sensed data, the scope of the present study is that it can test the validity of this multi-fractal-interslipface angle curves by studying the natural conical sand dunes which are predominant in Central Asia (e.g., Bharkan sand dunes).number of slipfaces (N= 2 in the present context, and the normalized fractal dimension (c0).Now this reduced equation is as follows.0 2 Sin -1 {[10 exp. (Log N/D)]/2} (3)   The computed fractal dimensions (D) and cor- responding interslipface angles (0) have been depicted in Table I.From the study, a multifractal and multi-interslipface angle curves are plotted

FIGURE
FIGURE 2aA multi-fractal curve of the simulated conical sand dune.