# Fluvial Transport Model from Spatial Distribution Analysis of Libyan Desert Glass Mass on the Great Sand Sea (Southwest Egypt): Clues to Primary Glass Distribution

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## Abstract

**:**

## 1. Introduction

## 2. Study Area

^{2}flat plain with numerous wadis; some of them, in the eastern flank of the plateau, conform to the ancient tributary system of the Gilf River. The northern part of this plateau is known as the Abu Ras Plateau, which is close to the LDG-strewn field. The Great Sand Sea is mostly composed of longitudinal sand dunes that extend for tens of kilometers; as high as 100 m, they have a quasi-north–south parallel arrangement. The northern depressions (Siwa and Qattara) are relics of fan deltas from the Oligocene (Figure 2), from the beginning of the Oligocene until the early-Miocene Epoch; the gradual uplift of southwest Egypt produced the regression of Tethys Sea northward, setting the shore to the Siwa or Qattara latitudes. The depressions were formed by the running water of the Gilf River that reached the shore, carving the depressions by dissolution of carbonates that were underlying the more resistant Miocene limestone dipping–northward-layers. This latter hypothesis on the formation of the northern depressions of the Western Desert of Egypt was supported by the discovery and study of paleorivers detected by radar, and it was formulated in conjunction with the fact that a humid climate during long time periods favored the recharge of the Gilf River from the highlands of the Gilf Kebir Plateau [4,35].

## 3. Methods of Geostatistical Study

#### 3.1. Sampling and Pattern of Univariate Data

^{2})). The source of every sampling point is explained in Table 1 of the Supplementary Material. The sampling design was not based on a grid or specified terrain divisions; thus, it was not straightforward to derive an original extension of the strewn field.

^{3}km

^{2}). The null hypothesis is a random Poisson process (H

_{0}: events exhibit complete spatial randomness), giving a modified exponential nearest neighbor distribution with mean:

**Figure 3.**The smallest rectangle enclosing the LDG sampling points has been drawn over the geographic map. The rectangular area (4.17 × 10

^{3}km

^{2}) was employed in the nearest neighbor analysis (R = 0.87), giving a clustered LDG point pattern.

**Figure 4.**The function for the observed LDG distribution appears in black; the three graphics correspond to functions K(d), L(d), and L(d)-d, which are above the 95% confidence interval (red functions), indicating a tendency towards a clustered distribution.

#### 3.2. Exploratory Statistics

^{2}) and the interquartile range (8.48 g/m

^{2}) for central tendency and dispersion, respectively. The value of the asymmetry coefficient, or skewness (3.3595), is markedly positive—that is characteristic of a logarithmic distribution. In a log-normal distribution (Figure 5a), high values increase the variance, which increases the difficulty of interpretation, calculation of the variogram, and the subsequent kriging interpolation. For this reason, we transformed the standardized variable Z(x)/A (g/m

^{2}) to Y(x) = ln[(Z(x) + 1)/A] (g/m

^{2}), where the constant 1 was introduced to avoid an indeterminate mass value (i.e., ln0). This transformed variable’s histogram suggested the existence of two modes (Figure 5b). Subsequent analysis of the mixture of populations was performed using the expectation–maximization (EM) algorithm [44] implemented in the PAST software package [43].

_{0}) is as follows: the sample was taken from a population with normal distribution. If the given p (normal) is less than 0.05, then normal distribution can be rejected. The results are summarized in Table 2. For the original distribution, as well as for its logarithmic transformation, the value of p (normal) is less than 0.05; however, when the subpopulations are considered, the values of p (normal) are above 0.05 and conditions of normality are accepted. In the case of the Shapiro–Wilk test, normality was accepted with a Wcrit (critical value of the Shapiro–Wilk test) of 0.90 and a probability—p (normal)—greater than 0.05. The Jarque–Bera method was in agreement with the results from the Shapiro–Wilk procedure.

**Figure 5.**(

**a**) Histogram of the variable g/m

^{2}of the LDG sampled points; (

**b**) Histogram of the logarithmically transformed variable.

Population | Probability | Mean | Standard Deviation |
---|---|---|---|

1 | 0.22055 | −5.6089 | 1.5233 |

2 | 0.77945 | −0.76634 | 1.4361 |

Normality Test | Original Distribution | Logarithmic-Transformed Distribution | Minor Mass Population | Major Mass Population |
---|---|---|---|---|

N (number of data) | 76 | 76 | 18 | 58 |

Shapiro–Wilk W | 0.6079 | 0.9411 | 0.9119 | 0.9795 |

p (normal) | 6.16 × 10^{−13} | 0.001534 | 0.09297 | 0.4313 |

Jarque–Bera (JB) | 327.5 | 7.586 | 1.186 | 1.49 |

p (normal) | 7.65× 10^{−72} | 0.02253 | 0.5528 | 0.4748 |

#### 3.3. Kriging Methods

_{i}) = value of log-transformed variable at spatial location (x

_{i}); and N(h) = number of pairs with separation distance h.

_{0}; $Y\left({{\displaystyle x}}_{i}\right)$ = sampling points; and ${{\displaystyle \text{\lambda}}}_{i}$ = weighting factor.

^{2}) by an adaptation of Yamamoto’s method [48]:

_{i}; and ${{\displaystyle Y}}^{*}\left({{\displaystyle x}}_{i}\right)$ = predicted value at the same location.

## 4. Digital Tools on Image Processing

- Preparation of the Digital Elevation Model (DEM).
- Elimination of the sinks. In order to obtain a “corrected DEM” for subsequent analysis, this function identifies depressed zones and fills them.
- Flux direction. The algorithm D8 traces the flow from every pixel of the DEM to one of the surrounding pixels. The result is the generation of flux lines that appear as lines perpendicular to the elevation contours.

- Determination of a raster with flow accumulation within each downslope cell. It is a calculation of the number of cells that flows into a particular downslope cell, according to flux direction [54].
- The paleodrainage network is a shape that results from the combination of the flux direction and accumulation layers by means of extension basin 1 [53].

## 5. Results and Discussion

#### 5.1. Geostatistical Mapping

^{2}). The corresponding map is illustrated in Figure 9, beside the distribution of the variable ln[(Z(x) + 1)/A] and the maps of the residuals (i.e., the true error of log-normal kriging).

**Figure 6.**Omni-directional semi

**-**variogram shows adjustment of the theoretical model to the experimental curve. Blue line is the mathematical semi-variogram model; the estimated semi-variogram is represented by the black line.

Parameter | Omnidirectional Semi-Variogram | Final Semi-Variogram |
---|---|---|

Nugget effect | 2.5 | 2 |

Sill | 9.0 | 17 |

Range (m) | 17,000 | 20,000 |

Nugget-sill-ratio | 0.28 | 0.12 |

Anisotropy direction (sexagesimal degrees) from E–W | - | 30° |

Direction of variogram (sexagesimal degrees) from E–W | - | 120° |

**Figure 9.**Contour maps of (

**a**) the log-normal kriging, established with a 150° directional semi-variogram; (

**b**) back-transformed variable with Yamamoto’s method; and (

**c**) the residuals or error quantification of the back-transformed variable.

^{2}, showing that the log-normal kriging is an unbiased estimator. The average magnitude of the forecast error, measured by the RMSE indices, equals 4.65 g/m

^{2}; it is worth mentioning that the RMSE is influenced more strongly by large errors than small ones.

#### 5.2. Understanding the Surface Distribution Model of LDG

**Figure 10.**Paleodrainage maps: (

**a**) Western Desert of Egypt; (

**b**) inset of the red square in a, where the red ellipse indicates a probable connection zone between the Kufrah and Gilf River basins.

**Figure 11.**Drainage pattern of the Kebira structure. Structural units of the Kebira were proposed in [23].

**Figure 12.**Map of the LDG distribution model. Red arrow: line of maximum gradient in LDG surface mass density. Purple arrow: line of LDG transport.

^{2}(Figure 13). In order to avoid overestimation of the mass of population 2, a “mixture” subpopulation associated to the more extended area (i.e., it is attributed to contours with intermediate values in Figure 13) is established (Table 4). Calculation of the total LDG mass resulted in a very low value compared with the simulation made by Artemieva et al. In the case of the Ries–Moldavite strewn field [58], they obtained the quantity of 1.6 × 10

^{13}g of tektites deposited in the simulation; on the other hand, Artemieva estimates a total mass of ejecta of ~ 4 × 10

^{15}g for the Australasian tektite strewn field [59], but for the LDG-strewn field Weeks et al. [9] calculated a mass of 1.4 × 10

^{9}g, while Barakat et al. [10] estimate a LDG mass equal to 2.67 × 10

^{8}g. Concerning these latter works, the result in Table 4 is very similar to the former work and almost one order of magnitude greater than the second. The occurrence of a layer of buried melted material is not ruled out in the Zone 3. In this sense, a radiative melting of target material from a low-altitude airburst is also compatible with the shape of the southernmost zone in the strewn field.

**Figure 13.**Relation between paleohydrography and contour map of LDG distribution over the Landsat-7 ETM+ reference map. The red dotted line delimits the most likely area associated to the melting event.

Population | Mean (g/m^{2}) | Approximated Area (m^{2}) | LDG Mass (g) |
---|---|---|---|

1 | 0.004 | 3.420 × 10^{8} | 1.368 × 10^{6} |

2 | 0.465 | 1.082 × 10^{9} | 5.031 × 10^{8} |

Mixture | 0.235 | 2.731 × 10^{9} | 6.404 × 10^{8} |

Total mass | 1.145 × 10^{9} |

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Jimenez-Martinez, N.; Ramirez, M.; Diaz-Hernandez, R.; Rodriguez-Gomez, G. Fluvial Transport Model from Spatial Distribution Analysis of Libyan Desert Glass Mass on the Great Sand Sea (Southwest Egypt): Clues to Primary Glass Distribution. *Geosciences* **2015**, *5*, 95-116.
https://doi.org/10.3390/geosciences5020095

**AMA Style**

Jimenez-Martinez N, Ramirez M, Diaz-Hernandez R, Rodriguez-Gomez G. Fluvial Transport Model from Spatial Distribution Analysis of Libyan Desert Glass Mass on the Great Sand Sea (Southwest Egypt): Clues to Primary Glass Distribution. *Geosciences*. 2015; 5(2):95-116.
https://doi.org/10.3390/geosciences5020095

**Chicago/Turabian Style**

Jimenez-Martinez, Nancy, Marius Ramirez, Raquel Diaz-Hernandez, and Gustavo Rodriguez-Gomez. 2015. "Fluvial Transport Model from Spatial Distribution Analysis of Libyan Desert Glass Mass on the Great Sand Sea (Southwest Egypt): Clues to Primary Glass Distribution" *Geosciences* 5, no. 2: 95-116.
https://doi.org/10.3390/geosciences5020095