Fractal dimension calculation of the geological formations and study of their relationship to the formation sensibility

Fractal analysis is one of the quantitative modeling of river networks. By determining the fractal dimension of linear structures such as faults, canals, and meandering river paths, one can estimate many of their features (Turcotte D.L. 1992). Fractal of figure, is a component with static geometric patterns that illustrates the general pattern of a phenomena. The first studies to create quantitative, mathematical, and geometric proper models from river network were made by Horton in 1932 and 1945, yet the study of relationship and comparison quantitative parameters with fractal geometry goes back to the last two decades.
2. Study area
The study area consist of 12 watersheds –Holeylan, Doyraj, Tangesazbon, kolm, NazarAbad, Jezman, Vargach, Chomgez, Chaviz, Siagav, JafarAbad, and Ema– from Ilam Province. Table 1 showed that the formation of study area.  
Table1 the details of formations in the study area.
Formation name Symbol Lithology Sensitivity to erosion of 10
Quaternary Qal Alluvial deposits of the platform 1
Quaternary Qt Alluvial fan 5
Aghajari Aj Sandstone – marl – sandy limestone - conglomerate 6
Gachsaran Gs Marl – limestone marl 3
Asmari Sb Karstic limestone - dolomite 9
Pabdeh Pd Mliky gray shale and marl with limestone 7
Kashkan Kn Conglomerate and sandstone and siltstone red 9
Ahak Tele Zang Tz The average white to cream-colored limestone marl layers 9
Amiran Am Siltstone and sandstone olive to dark brown color 7
Ahak Imam Ehm Rifi fossils of cream-colored limestone with interlayers of Chile 8
Sarvak Sr Thin layer of limestone 9
Ilam Il Medium to thin and milky gray limestone layer 7
In table (1) by increasing the numerical value of resistance degree, the formation sensitivity to erosion is reduced. (Rosovski van Voyk, 1992).
In FayzNiya’s classification (1995) which is based on Rosovski’s classification, rocks with greater resistance have higher value (max 20), and rocks with lesser resistance have lower value (min 1). Therefore, resistance range to erosion of the existing formations in the study areas can vary from 1 to 9.
3. Research Extraction of drainage network via Arc GIS
These networks were provided based on 50 DEM coordinates that in many cases, there isn’t enough accuracy and some channels are not displayed. Therefore, after transferring data to Google Earth, it was fully matched with the natural drainages and with a 5-meter accuracy, hydrographic network map was drawn and completed to reflect the full details of the network.
Thence one cannot scale maps via “Fractalys”, fields with the same space of 25 kilometers on similar formations in different areas were accidentally chosen via “Fish Net” –in Arc GIS, to fix this problem. For each study formation, three 25sq.km. Fields were chosen and by the accuracy of 5 meters. These maps that had the same drawing accuracy and space, were drawn in the same scales via GIS on an A4 page in “.bmp” and then were brought to Fractalys and finally, their fractal dimensions were calculated and extracted by the geometric method of counting boxes.
4. The The results show that a canal with an accuracy of 50 DEM meter on corresponding 5x5sq.km. plots, has much less accuracy than 5 meters in comparison to the drainage network drawn via Google Earth with less than a 5-meter accuracy, and in formations that are sensitive to the formations which are resistant to density changes of the hydrographic network, and have more changes in their fractal dimension as a result.
Google Earth images below are the examples of 25-kilometer areas which their hydrographic networks were revised.     Quaternaryn Gachsaran
Fig 6 modified hydrographic network of 25 km in Google Earth
Fig (6) Regression numeric index to erosion resistance (Sf) and formations fractal dimansion (Fr) after modification of 25 km units
In Fig (6) the amount of R2 is 0.9742 and shows the high correlation and significant relationship of fractal dimension to numerical index for resistance to erosion. by increasing the resistance of the formaion, numerical value of fractal dimension will be decreased.
Table 3 showed that statistical analysis between SF and FR.   Table 3 formations resistance data correlation (from 20) (Sf) and fractal number (Fr) of formations after correcting the 25-kilometer units in them.
Fr Sf
-965.0 000.0
12 1
12 Sf Pearson
Correlation
Sig. (2-tailed)
N
1
12 -965.0 000.0
12 Fr Pearson
Correlation
Sig. (2-tailed)
N
Table (3) shows the amount of data correlation (-965). Their amount would always be between the numbers +1 and -1. The more close its absolute gets to 1, the correlation will be higher, and the more close it gets to zero, the data correlation will get lower. (Ismayili and Kheyri 1385) As a result, there is a meaningful connection between formation resistant and fractal dimension. The minus sign indicates a negative data correlation (Afshani 1387).
Table 4 Regression of the formation resistance values (Sf) and fractal number (Fr) of the areas after the 25-kilometer unit correction
Model Summary
Std. Error of the Estimate Adjusted R Square R Square R Model
0.05116 0.924 0.931 0.965a 1
.
In the table above, “R” is the correlation intensity and its value are always between zero and +1 and “Square” is the coefficient of determination. The more closer “R” value gets to 1, the higher the correlation is between two variables. Therefore the number 0.965 illustrates high correlation of the formatuion resistance and its fractal number.
Total The results show that between the fractal dimension and hydrographic network, there is a significant and negative correlation. The highest amount of fractal dimension in study areas is for the Quaternary formation of granule, equals to 1.65, and the lowest numeral amount of fractal dimension belongs to “Sarvak” formation –equals to 1.06.
Also in formations with greater sensitivity than resistant formations after the correction of the hydrographic network via Google Earth, more changes occurred in the hydrographic network congestion, thereupon their fractal dimension change is also observerd more