ÿþ<HTML><HEAD><TITLE>GIS and Geostatistical Modeling of Surface Fractures and their Subsurface Extension : A Case Study in The Arabian Shield</TITLE> <META content="GIS and Geostatistical Modeling of Surface Fractures and their Subsurface Extension : A Case Study in The Arabian Shield" name=DESCRIPTION> <META content="GIS and Geostatistical Modeling of Surface Fractures and their Subsurface Extension : A Case Study in The Arabian Shield" name=KEYWORDS> <STYLE type=text/css> A:link {COLOR: 000080; TEXT-DECORATION: none} A:visited {COLOR: 000080; TEXT-DECORATION: none} A:active {COLOR: 000080; TEXT-DECORATION: none} A:hover {COLOR: red; TEXT-DECORATION: underline} </STYLE> </head> <body leftmargin="25" rightmargin="25"> <table border="0" bordercolor="#ffcc00" cellpadding="10" cellspacing="0" width="100%"> <tr> <td> <font face="arial" size="2"> <div align="justify"> <font face="verdana" size="1"> <b> GISdevelopment.net --> Proceedings --> GISDECO --> 2004 </b> </font> <br> <div align="right"> <input type=button value="Print" onClick="javascript:window.print()"> </div> <br> <font size=3 face=arial><b>GIS and Geostatistical Modeling of Surface Fractures and their Subsurface Extension : A Case Study in The Arabian Shield</b></font><br><br> <FONT face=verdana size=1> <b>Rao S. Divi</b><br> Dept of Earth & Environment, <br> Kuwait University, Kuwait </FONT><br><br> <STRONG>Abstract</STRONG><br><br> Tectonic fractures in rocks occur at all scales from microscopic to regional, and their spatial variations in density, length and orientation can be better understood by GIS data analysis and Geostatistical modeling. The GIS analysis includes data capture and query, and derivation of quantitative estimates of fracture characteristics. The Geostatistical modeling involves constructing experimental semivariogram for the fractures observed at specific locations in a region, fitting a suitable mathematical model to the variogram, and estimating the values of fracture characteristics at all locations in the region. The estimated values for the fracture characteristics on the surface can be considered to continue to the same degree into the subsurface. As an example, fractures in the Precambrian Arabian Shield rocks near Abha, Saudi Arabia are used in such modeling. This approach has application Engineering and hydrogeological geological studies. <br><br> <STRONG>Key words</STRONG><br><br> GIS, Geostatistics, Fractures, Arabian Shield <br><br> <STRONG>Introduction</STRONG><br><br> Spatial distribution of tectonic fractures in rocks depend on many geological factors including degree of deformation, lithological variations, scale of observation etc. These, together with the common constraint of limited available outcrops for observation of fractures, result in complex spatial patterns. Understanding such patterns needs quantitative data analysis and modeling, and in the recent times, GIS and geostatistical techniques have been widely used in such analysis and modeling (Isaaks and Srivastava 1989, Cressie 1993, Goovaerts 1997). The objective in these investigations is to define an appropriate spatial model (Variogram) for the distribution of observed fracture characteristics (e.g.size) at specific locations in a given region, and to estimate (Kriging) the characteristics at other locations in the region. <br><br> <STRONG>Variogram and kriging</STRONG><br><br> The variogram is the basic geostatistical tool for measuring spatial autocorrelation of a regionalized variable. For a spatial variable z, changes in its value between sample-pairs at location x and location x+h (h-units away from x) are measured as the differences z(x)  z(x+h). If the surface represented by the two points is continuous and h is a small distance, the differences will be small. With increasing h, the differences increase. For n observations in a region of study, the variogram ³(h)* is computed as follows : <br><br> <img src=images/rao1.jpg border=0> <br><br> (1) The next step involves fitting a theoretical model ³(h) for the observed variogram ³(h)*. The three features of observed variogram that guide in fitting a theoretical model are : (1) presence or absence of sill C, which is indicated by the leveling off of the variogram once h increases beyond some distance(range) a, and (2) behavior(shape) of the variogram at the origin, and (3) presence of absence of nugget effect C0, indicated by an interceptof the variogram on the y-axis of ³(h)*-h graph. The nugget effect implies abrupt changes in the regionalized variable over small distances, variability at a spatial scales finer than sample spacing. The three models that are commonly used are : spherical, exponential and Guassian. The spherical model, by far the one most often used in recent years, is given by : <br><br> <img src=images/rao2.jpg border=0> <br><br> Kriging is a method of calculating estimates of a regionalized variable at a point, over the region of study, and uses as a criterion the minimization of an estimation variance. Calculated at intersections of a regular grid, kriged estimates can be used for drawing a contour map. A kriged estimator zk* is a linear combination of n values of the variable : <br><br> <img src=images/rao3.jpg border=0> <br><br> (4) where »i are weights calculated according to the criteria that the estimate is unbiased and the estimation variance is minimized. <br><br> <STRONG>The case study</STRONG><br><br> <STRONG>The data</STRONG><br><br> The data used are the tectonic fractures in the Precambrian rocks of the Arabian Shield in the Abha region of Saudi Arabia . Topographic variations in the region are high, with very low elevations in the coastal areas near the Red Sea to the west, a transition zone with very steep contours reflecting rapid elevation increase (3000 meters) towards west, and a plateau with undulating topography in the west. Because of this characteristic physiography, rainfall is frequent with flash floods in the coastal areas, and construction of dams at various locations are planned to control the floods. For the selection of optimal locations for dam construction, one of the requirements is to investigate into the  degree of fracturing in the bed rocks on which the dams are to be constructed. However, fracture development in the rocks is related to complex geological factors that include variations in tectonic deformation, nature of rocks, and varying scales (microscopic to regional) scales of fracture development. As such, geologically mapped fracture pattern is spatially inhomogeneous, and the analysis needed some objective and rigorous statistical analysis of the data. The fractures in the region are mapped from aerial photographs and satellite images, and ground checked in limited areas (Fig. 1; Divi et al, 1996). Fig. 1 : Fracture map of Abha area <STRONG>Analysis</STRONG><br><br> The fracture map is analyzed under GIS facility to derive fracture parameters, such as orientation, length, spacing (number of fractures in a unit area) etc. of the fractures. These parameters are used to construct variograms and derive kriging estimates. In this study, spherical model is fitted to the experimental variogram. From the fracture map, the number and total length of fractures per unit square cell (8.5km x 8.5 km) are calculated . These data are used to construct the experimental variogram and contour map of kriged estimates (Figs. 2) <br><br> Fig. 2 : Contour map of Kriged estimates of number of fractures in unit cells. <br><br> The complex spatial distribution of the fractures in Figure 2 is due to (1) widely varying lithologies that include very rigid and strong granites and granitic gneisses, moderately strong quartzo-feldspathic and mafic gneisses and weak mica-rich schists; (2) inhomogeneous strain in the rocks, as reflected by discrete and discontinuous shear zones with high strain,; and (3) discontinuous nature of outcrops because of desert sand cover. However, it is believed that the delineated fractures represent a sample of the true population of the fractures in the rocks in the region. As is evident, the fractures extend for different lengths ranging from a few meters to more than thirty kilometers, densities and orientations. In this study fractures of all orientations are included in the analysis, because, the  degree of fracturing is the factor that is critical to rock strength in dam construction. Since outliers have undue influence on the shape of experimental variogram, frequency histograms are used to identify and discard the few extreme values. The experimental variograms constructed for the number and total length of fractures in unit cells suggest absence of nugget effect and that spherical model is more appropriate to be fitted to the data. The variogram for the number of fractures in unit cells indicate a sill value of 20 and a range value of 30 kilometers, after which the graph begins to flatten out. For the total length of fractures, the sill value is twenty and the range value is 28 kilometers. These values are used to calculate the estimates for other locations by kriging <br><br> The contour map in Figure 2 points out to the areas of high fracture intensity in the rocks. For example, in the bottom right hand corner, both the figures indicate high intensity reflecting the general relation between fracture density and length. However, to the northwest of this area along the Red Sea, fracture intensities in the two maps differ suggesting lack of such relationship. Regardless, the two maps are useful in aiding in dam site planning. Field verification and further analysis will refine the preliminary results in this study. <br><br> <STRONG>References</STRONG><br><br> Cressie, N.A.C.. (1993). Statistics for spatial data. John Wiley & Sons, New York. <br><br> Divi, S.R., Chagarlamudi, P, Zakir, F.A. and Al-Wash, M.A. (1996). Procedures for enhancement, delineation and analysis of structures from satellite images, King Abdulaziz City for Science and Technology (KACST), Riyadh, Saudi Arabia, Technical Report TR-4/ST 691. <br><br> Groovaerts, P. (1997). Geostatistics for natural resources evaluation, Oxford University Press, Oxford. <br><br>Isaaka, E.H. and Srivastava, R.M. (1989). An introduction to applied geostatistics, Oxfor University Press, Oxford. <br><br> </div> </font> </td></tr> <tr><td align=center><font face=verdana size=1 ><b>© GISdevelopment.net. All rights reserved.</b></font> </td></tr> </table> </body> </html>