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Interactive spatial analysis of lineaments
Computers & Geosciences 36 (2010) 1081–1090
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Interactive spatial analysis of lineaments$ Thushan Chandrasiri Ekneligoda a,b, Herbert Henkel a,n a b
Department of Land and Water Resources Engineering, Royal Institute of Technology, Stockholm S-100 44, Sweden Itasca Geomekanik AB, Stockholm, Sweden
a r t i c l e in fo
abstract
Article history: Received 28 November 2009 Received in revised form 27 January 2010 Accepted 30 January 2010
An interactive software tool, here called Spatial Analysis of Lineaments (SAL), has been developed for calculating the spatial properties azimuth, length, spacing, and unidirectional frequency of lineaments which are defined by their start and end coordinates. In a series of steps the user is guided by displays of relevant statistical distributions, which can be user designed. Statistical outliers can be excluded and the total sample of lineaments can be subdivided into azimuth sets and, if required, into spatial clusters. Special attention is given to the removal of spatial outliers in an interactive way. Several rule-based decisions are made to determine the nearest lineament in the spacing calculation. As a default procedure, the program defines a window whose size depends on the mode value of the length distribution of the lineaments in the study area. The software can accept a large amount of lineaments and can analyze the spatial properties of each azimuth set avoiding the repetitive calling of the original database. A simple rule was developed to derive the unidirectional lineament frequency. The spatial properties are presented as histograms for each azimuth set together with the mode, mean, standard deviation, and number of involved lineaments. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Spatial properties Azimuth Length Spacing Spatial outliers Uni-directional lineament frequency
1. Introduction The study of spatial properties of lineaments has attracted research interest in the past because it can provide valuable information about natural resources as well as hazard assessment. Linear features can possibly represent a discontinuity in a mineral, in the rock mass, or in the earth’s crust. The length of these can vary over a wide range of 10 orders of magnitude from 105 to 10 5 m (i.e. from continental scale to mineral scale). In this study, linear features are for simplicity categorized as ‘‘lineaments’’. Lineaments related to geological displacement zones (faults and thrusts) can be described in a simplified way as illustrated in (Fig. 1). The linear trace in any data set is thus related to high angle dips of the displacement zone. A different spatial resolution of the data set studied also affects what can be classified as lineaments. The anastomosing shear zone pattern in the example in (Fig. 1) could thus appear as a linear feature in a dataset with lower spatial resolution. The spatial properties of lineaments can be used in several fields of geosciences. In rock engineering, spatial properties of
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Code available from server at: http://www.iamg.org/CGEditor/index.htm. Corresponding author. Fax: + 468 790 68 10. E-mail addresses: [email protected], [email protected] (T. Chandrasiri Ekneligoda), [email protected] (H. Henkel). n
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lineaments are important in order to characterize the rock mass geometry and finally the rock mass properties. Hudson and Priest (1979, 1983, 1976) studied rock joint distributions in detail and tried to relate the discontinuity frequency in an image with the rock quality designation (RQD) value. They attempted to fit the spacing distribution approximately to a Poisson distribution and used a model to relate this approximated distribution to the RQD value. Studies of lineaments to locate geological hazard zones have been carried out by some researchers. In Koike et al. (1995) and Koike and Ichikwa (2006), lineaments that appear in satellite images can be related to tectonic features. Spatial correlation of lineaments from remote sensing data, borehole fracture data, and micro-cracks in minerals, has been used to derive a scaling law. Several researchers have attempted to relate lineaments and ground water resources (Mabee et al., 1994; Lattman and Parizek, 1964). According to Kim et al. (2004) the occurrence of lineaments has a close link to ground water prospecting. In such applications the number of lineaments that can be identified in a specified area has an assumed relationship to ground water productivity. Previous studies of lineament spacing based on geophysical data are reported e.g. in Henkel (1979) for magnetically indicated fracture and fault zones, and in Henkel (1992) for VLF (Very Low Frequency) indicated low conductivity zones, or for topographic escarpments (Henkel, 1992; Henkel et al., 2005). VLF data are sensitive to contrasts in electric conductivity in soil and bedrock. In regions with thin soil cover and underlying high resistive
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Fig. 1. Lineaments and displacements. Steeply inclined faults (like normal faults) result in straight traces (in geophysical or morphological data) enclosing angular blocks. Inclined faults (thrusts) result in curved lines. Steeply inclined shear zones may produce anastomosing traces enclosing shear lenses.
crystalline bedrock, conductive features such as steeply inclined fracture zones in the underlying bedrock can be mapped with VLF measurements (Oskooi, 2004), and are commonly approximated by straight lines (Henkel, 1992). Lineaments and fractures usually have complex azimuth distributions, as in the example given in (Fig. 2a and b), which is based on the interpretation of magnetic anomalies (Henkel, 1979). We also found in the analysis of VLF data that three to four clearly separable azimuth sets can be identified in one area. A case as illustrated in (Fig. 2b) requires an advanced tool for the separation of azimuth sets. Lineament studies are carried out in three steps, i.e., lineament extraction, derivation of lineament properties and their interpretation in a geological (or other) context. Lineament extraction is a challenging area and several authors have provided different tools. Some have attempted to extract lineaments using computer tools (e.g. Burdick and Speirer, 1980; Raghavan et al., 1995). The aim of this paper is to deal with the second of the three steps. For the derivation of spatial properties of lineaments, several approaches can be made. Two methods are presented in Clark and Wilson (1994). One method allows the user to handle a large number of data sets and is generally applied to satellite images. The sampling area can either be a square or a rectangular grid (Kumar and Reddy, 1991), or circular (Mostafa and Qari, 1995). In the other approach each lineament is considered and the relations to its neighbors are derived. Clark and Wilson (1994) developed software to derive the spatial properties of parallel lineaments for such an approach. It requires parallel lineament sets defined by end coordinates and mid coordinates. A computer program in C language derives the spatial properties length, orientation, and parallel spacing. The lineament density can be expressed in three ways: as the number of lineaments per unit area, as the total length of lineaments per unit area, or as the number of cross points per unit area (Mostafa and Qari, 1995; Kim et al., 2004). The results of those methods that derive new indirect measures of spatial properties are intrinsically difficult to interpret in terms of the expected geological concepts like faults and fractures. The increased coverage with detailed aero-geophysical data and the lack of relevant tools to derive spatial properties which are representative for a study area,
encouraged the design of a software tool to determine the spatial properties of lineaments more systematically. The first version of our program was tested and validated on lineaments derived from low altitude airborne Very Low Frequency (VLF) electromagnetic measurements. Although we use some of our previous modules in Ekneligoda (2004) this version extends the analysis in several ways in order to improve the results and provide an enhanced method to carry out lineament studies. The new version allows the user to select the appropriate azimuth set in an interactive way and provides a facility to analyze the other azimuth sets subsequently. Statistical outliers can either be excluded or, in the case of spatial outliers, be treated as individual sets. The proposed Spatial Analysis of Lineaments (SAL) tool can be used for any set of lines and therefore for a range of various applications. All lineaments are included and treated individually to derive statistical distributions of the spatial properties length, azimuth, spacing and the spatial (2-D) frequency. The characteristic statistical parameters average, mode, and standard deviation are calculated for azimuth, length, and spacing. In the next section the basic definitions applied in the software tool are explained. This is followed by a description of the methods developed for the analysis, with emphasis on the spacing and spatial frequency calculation, and then the different steps of an analysis are described with reference to the display windows where the user can interactively design the analysis.
2. Definitions 2.1. Map area The study area is specified by giving the lower left and upper right corners in planar coordinates. The user can select any type of units, but it is recommended to use SI units. 2.2. Lineament The term lineament is used to cover any type of linear feature in any kind of data. These could be traces of discontinuities in a
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distribution, which can be representatively described by the mean value and the variance. The observed distribution may be skewed or contain a few members with a non-characteristic azimuth. Such azimuth outliers can be analyzed as a separate set. 2.4. Length The lineament length is the distance between the start and end coordinates, scaled to a systematic unit (preferably in m). The length distribution for given azimuth set is in theory exponentially decreasing and would contain values from 0 to infinity. In practice, the lineament length is restricted to the smallest spatial resolution (like the spacing between flight lines) at the low end, and to the diagonal of the study area at the high end. In this case the distribution is best represented by the mode value. The distribution may contain atypical members, length outliers, for which the user has to decide if they should be included or excluded in the analysis. 2.5. Spacing The spacing is the perpendicular distance to the nearest lineament of the same azimuth set, and is scaled to the same unit as length. The spacing between two lineaments is not constant throughout their lengths. Since the lineaments usually are not exactly parallel, the spacing is calculated for lineaments within a restricted azimuth range. The spacing distribution depends on the configuration of the search window which the user has to define. With a large search window, the spacing distribution tends to be bi-modal with an increase of the number of multiples of the first mode. This issue is further discussed in the next section. 2.6. Lineament occupancy area A user-defined circumscribing polygon, that covers the lineament set studied, is the lineament occupancy area. The rules for the definition of the construction of the circumscribing polygon are given in the methods section. It is quite common that some lineaments of an azimuth set are located outside their main spatial cluster as spatial outliers (Fig. 3). In connection with the spacing calculation, such spatial outliers should be excluded. If the lineament frequency for several direction sets within an area is required, a separate determination of the common polygon for all lineament sets has to be made. The spatial frequency of lineaments, (or lineament density) is defined here as the two-dimensional frequency, i.e., the total length of a one or several lineament sets divided by the area it Fig. 2. (a) Example of a traditional lineament interpretation based on aeromagnetic data in a 250 500 km region in Northern Sweden (from Henkel, 1979, numbers and letters along frame denote national map sheet identifications). (b) Corresponding azimuth distribution. (Vertical axis shows frequency in %, and horizontal the azimuth in degrees, respectively.)
rock mass or in the earth’s crust. Lineaments are approximated as straight lines and are represented by their planar start and end coordinates, respectively, as seen in the left part of (Figs. 1 and 2). 2.3. Azimuth The azimuth is defined as the angle between a lineament and the grid north and can vary within a 1801 interval. The azimuth distribution of lineaments in a given map area may contain several azimuth sets of lineaments with similar azimuth as in (Fig. 2b). The azimuth distribution for a single set is considered to be a normal
Fig. 3. Examples of spatial outliers (dark gray and outside polygon encircling main spatial cluster).
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occupies. The unit is thus [length]/[length]2, i.e., m number of lineaments per unit area.
1
and not the
3. Methods 3.1. Visualization methods The interactive SAL software is based on display windows where the user can set display parameters and enter decisions about parameters that control the subsequent steps in the analysis. Updates of diagrams due to changed visualization parameters are linked and presented automatically. In the displayed diagrams, map frames and axes are displayed in black, lineaments and histograms in red, and added information (like polygons or mode values) in blue. The results from all calculations that concern spatial properties are displayed as histograms, together with the information about the statistical properties average, mode, standard deviation, and number of lineaments. Special attention was therefore given to improve the design of histograms. The frequency scale of histograms can be designed by the user, who can choose between the ranges 20, 50, or 100%. A variable minor subdivision, coupled to the major, is also provided to read off the intermediate values. The distribution of azimuths is usually symmetric and the mean is therefore considered to be representative. The distribution of lengths is however commonly positively skewed and preferably represented by the mode value. The distribution of spacing has a tendency to become bimodal when large search windows are used. In that case the first mode is selected as the mode value. The mode value is calculated as the mid value of the class that has the highest frequency. It is thus dependant on the selected class width. To illustrate the azimuth distribution, three different class widths are provided viz. 21, 51, and 101. The distribution is illustrated both in Cartesian coordinates as a histogram and in polar coordinates as a rose diagram (Henkel, 1968). These two ways of visualising are linked so that the rose diagram is updated when the histogram settings are changed. The range of the length and spacing histograms can be set individually to three different values: 1000, 2000, or 5000 units. This provides a better resolution adapted to the actual ranges of length and spacing.
3.2. Lineament orientation This procedure starts by checking the lineaments for their orientation and switches, when necessary, the start and end coordinates into the required order.
3.3. Lineament length adjustment For measurements that are arranged in a rectangular grid, the length of lineaments is reduced with respect to the measurement spacing and the azimuth of the lineament set, (Fig. 4). This modification is made for individual azimuth sets and the user has to determine an appropriate spatial resolution number.
3.4. Azimuth re-assembling In cases where azimuths occur outside the interval 0–1801, a routine is built to re-arrange the distribution into a continuous histogram for the interval 0–1801.
Fig. 4. Interpreted length of lineaments is reduced with spatial resolution, which may be different in different directions, i.e., depending on angle v to measurement grid. (Figure shows situation for airborne measurements where measurement spacing is smaller along flight lines and not necessarily aligned to a straight line.)
3.5. Separation of azimuth sets If several sets of azimuths are encountered, the selection of one azimuth set is required to continue the analysis. The user can define the lower and upper values of the azimuth angle based on the frequency minima occurring in the azimuth distribution. The accuracy of determining the lower and upper value can be enhanced by selecting the smallest class width (Fig. 5).
3.6. Calculation of spacing The spacing between two lineaments is calculated by creating roughly equi-dimensional blocks between the two lineaments. A typical variation of spacing is illustrated in Fig. 1 in the paper by Ekneligoda (2004). The program first calculates the perpendicular distance (d1) between the current lineament and its neighbor, then it goes forward the distance d1 along the current lineament, and draws the second perpendicular projection to the neighbor line (d2). Next, it moves the distance d2 along the current lineament to draw the third perpendicular (d3) projection. This procedure continues until the lineament is not long enough to match the extent of (dn). The selection of several distance calculations between pairs of lineaments gives a better spatial representation of the spacing distribution, as the calculations are spread over the entire area where the set of lineaments occurs. How spacing is calculated, illustrated for a strictly parallel set of lines (in a real case all directions within one azimuth set are considered), is presented in (Fig. 6). The current lineament is shown with the thicker line with a rectangular node. The same set of lineaments is shown with three examples of spacings calculated (or refused) related to a chosen current lineament. The non-resulting test for a subsequent spacing is only illustrated for the upper left case in (Fig. 6). The search window is the area on each side of a lineament within which the distance to a neighbor lineament is calculated, as illustrated in (Fig. 6). To include all possible spacings between neighbor lineaments, a front window and a back window is created, based on the length distribution of a set of lineaments. As lineaments can occur in different lengths and with different orientations, the use of both windows is required, as otherwise there is a risk of losing candidate neighbor lineaments. SAL provides the facility to vary the size of the search window in the off lineament (perpendicular) direction by multiplying the
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Fig. 5. Example of separation of azimuth sets by using a smaller class width. (Vertical axis shows frequency in %, and horizontal the azimuth in degrees, respectively.)
Fig. 6. How spacing is calculated is illustrated for strictly parallel set of lines (in a real case all directions within one azimuth set are considered). Same set of lineaments is shown with 3 examples of spacings calculated (or refused) related to a chosen selected lineament. Non-resulting test for a subsequent spacing is only illustrated for upper left case. User-defined size of calculation window is critical as it defines which of the neighbor lineaments are considered for a spacing calculation.
mode of the length distribution with a factor 1, 1.5, or 2 to find the nearest lineament. Different window sizes can be tested to find the optimal coefficient. For this purpose, a map is provided
containing the lineament arrangement and the first calculated spacing (i.e., d1 in (Fig. 1) in Ekneligoda and Henkel (2006)) for each pair of lineaments.
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3.7. The spatial occupancy polygon The polygon is drawn interactively by joining the endpoints of the outer lineaments of a spatial cluster as illustrated in (Fig. 3). The user-defined polygon is used to remove the spatial outliers and for the definition of the area used for the spatial frequency calculation. The user can make any complex shaped polygon to exclude empty areas. Lineaments that are cut by the polygon are included in the subsequent calculation. The user-defined polygon is expanded to get a more representative area. This concept is connected to the common occurrence of gaps in the spatial distribution of lineaments. As a provisional approach, the expansion is related to both the length and spacing statistical properties. In the lineament direction the expansion is based on the length statistical properties, whereas in the perpendicular direction the expansion is based on the spacing statistical properties. This complex expansion criterion is simplified by taking the average value of both length and spacing mode values. The expanded polygon is displayed in (Fig. 10) together with the result of the frequency calculation.
4. Spatial analysis of lineaments (SAL)
Fig. 7. Work path for lineament analysis using SAL tool.
The SAL software was first built using Visual Basic 6, which is considered to be a user-friendly computer language partly due to its advanced graphical interface (Peasley and Simon, 1998; Perry, 1999; Foxall, 2002). The new extended program consists of five display windows and several modules. The work flow chart of the extended program is presented in (Fig. 7).
Fig. 8. Display window 2. Azimuth distributions with several lineament sets, spatial outliers and rose diagram are shown. (Vertical axis shows frequency in %, and horizontal the azimuth in degrees, respectively.)
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Fig. 9. Display window 3. Histogram and statistics of azimuth (left, in degrees) and length (right, in m) of lineaments when spatial outliers have been removed. (Vertical axis displays frequency in %.)
In the first display window, the user is asked for the location of the database, which contains the starting and ending coordinates of the lineaments. The user can move to the second display window by clicking the button provided in the lower right portion of the window. In this display window the azimuth is shown in the histogram form and as a rose diagram and a map view of all lineaments is presented in (Fig. 8). In this window the polygon for the lineament density determination has to be defined. Once the azimuth set is selected, a map view of its lineaments is provided next to the azimuth distribution. At this point, the user can move to the third display window, where the compilation of length and azimuth is presented in (Fig. 9) for the selected set after excluding spatial outliers. Two histograms are provided for the interaction with the user. The scales of the histograms can be adjusted according to the range of each property azimuth, length, and the frequency, respectively. The histogram parameters’ range and class width are also displayed. The save button can be used to save the statistical values of length and azimuth. The spatial frequency (or lineament density) calculation is made in the same window. This procedure requires the definition of the area that is occupied by the lineaments that is studied. In most situations this area has an irregular outline, as seen in (Fig. 3). The user has to define this area as the polygon that connects the outer ends of the lineaments. Then the user can move to the fourth display window (Fig. 10) where the spatial arrangement of the lineaments is illustrated. The first spacing calculations, for each lineament are shown with different colours for back and front window. The list box shows the coordinates between which the spacings were calculated, and the corresponding spacing values. As the expansion length of the polygon is based on the average value of length mode and the spacing mode, the facility has been provided to add the length
histogram on the spacing histogram at this level. Each mode value together with their average value is also drawn with the corresponding color. The save button can be used to save the statistical values of spacing and lineament frequency. In the final display window (Fig. 11), the histograms and the statistical properties: mean, standard deviation, mode, and number of lineaments are summarized for each spatial property. In addition to azimuth, length and spacing histograms a rose diagram of the azimuth distribution and the user-defined polygon for lineament density calculation is also presented. After analyzing an azimuth set, the user can go to the next set by using the ‘Next azimuth set’ command. This procedure is repeated for all the azimuth sets without closing the program. The new program allows the user to handle about 30–40 azimuth sets though in reality maximum number of azimuth sets may be less than 10. The save button can be used to save all statistical values with all spatial properties.
5. Discussion The spatial precision of the entered lineament coordinates depends on the spatial resolution of the original data from which the lineaments have been derived. The pointing uncertainty in connection with the on-screen digitizing is accounted for by reducing the lineament length with the spatial resolution. Airborne measurements in particular have a strongly anisotropic sampling pattern, which may require a more complex approach of the spatial resolution, as it is different in different directions and hence affects the lineament length to a variable degree. Therefore it is recommended that the user has to find a suitable reduction measure to each individual direction set of lineaments.
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Fig. 10. Display window 4. Histogram of spacing and length (in m) and a map of first set of spacings in gray tones (in software green within front window, red within back window) together with lineaments considered in analysis. It also displays user-defined polygon and expanded polygon. (Vertical axis displays frequency in %.)
The common situation is that all lineaments appear together in one map. In some cases a separation into different azimuth sets is not obvious as lineaments azimuth angles can differ by few degrees only. It is always advisable to use a histogram of the azimuth for the separation of different azimuth sets. In our new tool we have provided an enhanced facility in order to carry out an intelligent azimuth separation. Some lineaments can appear away from the main cluster (spatial outliers). These outliers may not contribute to any spacing values in the subsequent stages as they are too far away from their neighbors. In such a case it is appropriate to remove them from the set before the subsequent statistical analysis is made. Our tool provides a method to remove these spatial outliers interactively. The user can identify the outliers at the beginning as the map view is provided next to the azimuth distribution. After drawing a polygon in an appropriate way the user can take away the outliers just by using the outlier exclusion button. This way the representativity of the statistical properties of azimuth, length, spacing, and lineament density has been increased. If the user is interested in getting the information about azimuth and length outliers, this information can be collected by selecting the corresponding azimuth interval. In the design of diagrams showing results, the user has several choices to find a suitable scaling of the histograms. The calculation of the mode value as the mid value of the class with the highest frequency is valid, when the right and left side frequencies are the same. In other cases a negligible error is admitted. The spacing calculation can be carried out by using different sizes of the search window. The user can choose an appropriate
factor. The use of a higher factor adds longer spacing values by connecting more lineaments thus contributing the right tail of the spacing histogram. This may lead to bi-modal spacing distribution. The use of unit factor on the other hand significantly cuts off the higher spacing values and reduces the cluster of lineaments for the subsequent analysis. The statistical representativity of a derived property for an area depends on a representative sampling scheme. An (at least roughly) equi-dimensionally spaced sampling should thus be attempted. That is the reason to design the sampling of spacing values at distances that are comparable to the spacing itself. The lineament density is derived based on a circumscribing polygon defined by the user (Fig. 3). This polygon should not include empty areas and be drawn close to the end of lineaments. Lineaments that are cut by the polygon will be included in the calculations. Considering the uncertainty of the gap length, the polygon construction does not require a painstaking precision (just round the lineaments) and if required, a test with differently drawn polygons may reveal the influence on the lineament frequency. The representative area of a set of lineaments reaches outside the lineaments. In the direction perpendicular to the strike this would correspond to one half spacing distance. In the direction parallel to the lineaments it would correspond to half the length of the gaps between lineaments, i.e. the distance between perpendicular lines at the start and end of neighboring lineaments. As the distribution of the gap length may be similar to the distribution of length, we have approximated the expansion length in the strike direction with the mode of the length
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Fig. 11. Summary of results uni-directional azimuth (histogram and rose diagram), length and spacing distribution together with map view of lineaments and polygons involved in lineament frequency calculation are presented. Also, unidirectional lineament density is shown.
distribution. To expand the polygon to the representative area, a compromise solution is given by using the average of the mode value of the length and the spacing distribution. We believe our new compromise solution results in a more representative area as it takes into account both length and spacing simultaneously. In case the area is a window out of a larger continuously represented region, the area of the window itself should be used for the calculation of the spatial frequency. If the studied region contains several spatial clusters of lineaments, it is advisable to subdivide the lineaments into several spatial sets in order to comply with the requirements for representativity of derived properties. A similar reasoning can be made regarding empty areas and a suitable subdivision may reduce the under-sampling effect that otherwise would be caused. Another factor of importance is the dimension range within which the sampling is made. Lineaments that represent fracture lengths are usually part of a fractal continuity ranging over many orders of magnitude (Gillespie et al., 1993) The observed dimension range is restricted by the area dimensions (the spatial coverage) of the surveyed region on the high end, and by the spatial resolution of the measurements on the low end. The distribution of fracture length gives an indication of the presence of one or several categories with different characteristic lengths. In a further development of the spatial analysis of lineaments, statistics about the gap length can be calculated and used for a more precise determination of the occupancy area. The multiple fracture frequency, based on a common polygon for all azimuth sets is under construction. In the case of shear zone geometry of lineaments, additional modules can be designed to describe other
relevant spatial properties like shear lens shape and shear lens dimensions.
6. Conclusions The presented tool for spatial analysis of lineaments (SAL), is a versatile interactive software tool for detailed characterization of lineaments based on the statistical criteria. The tool can handle a large amount of any line features at any scale and presents the results of an analysis in quantitative properties that are spatially and statistically representative. The spatial properties of different azimuth sets are derived in sequence, thus avoiding the repetitive calling and opening of the original database. Statistical outliers of the spatial properties can be removed prior to the calculations which improve the representativity of derived results. If spatial outliers cluster in some part of the study area, they can be analyzed as a separate group by a mutually excluding procedure. Complex azimuth distributions can be separated interactively into individual azimuth sets in a logical way with the zooming function by which the class width can be reduced. SAL provides a compromise solution for the determination of the polygon for the lineament density calculation by taking the average value of the length and spacing mode as the expansion length. A summary of results is provided in graphical form for presentation and comparison purposes. Our software would be useful in building a fully automated interactive analysis tool starting from the lineament extraction level. The software shows a path to also analyze shear
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zone related features, provided the spatial description of shear zones can be given in planar coordinates.
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Interactive spatial analysis of lineaments$ Thushan Chandrasiri Ekneligoda a,b, Herbert Henkel a,n a b
Department of Land and Water Resources Engineering, Royal Institute of Technology, Stockholm S-100 44, Sweden Itasca Geomekanik AB, Stockholm, Sweden
a r t i c l e in fo
abstract
Article history: Received 28 November 2009 Received in revised form 27 January 2010 Accepted 30 January 2010
An interactive software tool, here called Spatial Analysis of Lineaments (SAL), has been developed for calculating the spatial properties azimuth, length, spacing, and unidirectional frequency of lineaments which are defined by their start and end coordinates. In a series of steps the user is guided by displays of relevant statistical distributions, which can be user designed. Statistical outliers can be excluded and the total sample of lineaments can be subdivided into azimuth sets and, if required, into spatial clusters. Special attention is given to the removal of spatial outliers in an interactive way. Several rule-based decisions are made to determine the nearest lineament in the spacing calculation. As a default procedure, the program defines a window whose size depends on the mode value of the length distribution of the lineaments in the study area. The software can accept a large amount of lineaments and can analyze the spatial properties of each azimuth set avoiding the repetitive calling of the original database. A simple rule was developed to derive the unidirectional lineament frequency. The spatial properties are presented as histograms for each azimuth set together with the mode, mean, standard deviation, and number of involved lineaments. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Spatial properties Azimuth Length Spacing Spatial outliers Uni-directional lineament frequency
1. Introduction The study of spatial properties of lineaments has attracted research interest in the past because it can provide valuable information about natural resources as well as hazard assessment. Linear features can possibly represent a discontinuity in a mineral, in the rock mass, or in the earth’s crust. The length of these can vary over a wide range of 10 orders of magnitude from 105 to 10 5 m (i.e. from continental scale to mineral scale). In this study, linear features are for simplicity categorized as ‘‘lineaments’’. Lineaments related to geological displacement zones (faults and thrusts) can be described in a simplified way as illustrated in (Fig. 1). The linear trace in any data set is thus related to high angle dips of the displacement zone. A different spatial resolution of the data set studied also affects what can be classified as lineaments. The anastomosing shear zone pattern in the example in (Fig. 1) could thus appear as a linear feature in a dataset with lower spatial resolution. The spatial properties of lineaments can be used in several fields of geosciences. In rock engineering, spatial properties of
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Code available from server at: http://www.iamg.org/CGEditor/index.htm. Corresponding author. Fax: + 468 790 68 10. E-mail addresses: [email protected], [email protected] (T. Chandrasiri Ekneligoda), [email protected] (H. Henkel). n
0098-3004/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cageo.2010.01.009
lineaments are important in order to characterize the rock mass geometry and finally the rock mass properties. Hudson and Priest (1979, 1983, 1976) studied rock joint distributions in detail and tried to relate the discontinuity frequency in an image with the rock quality designation (RQD) value. They attempted to fit the spacing distribution approximately to a Poisson distribution and used a model to relate this approximated distribution to the RQD value. Studies of lineaments to locate geological hazard zones have been carried out by some researchers. In Koike et al. (1995) and Koike and Ichikwa (2006), lineaments that appear in satellite images can be related to tectonic features. Spatial correlation of lineaments from remote sensing data, borehole fracture data, and micro-cracks in minerals, has been used to derive a scaling law. Several researchers have attempted to relate lineaments and ground water resources (Mabee et al., 1994; Lattman and Parizek, 1964). According to Kim et al. (2004) the occurrence of lineaments has a close link to ground water prospecting. In such applications the number of lineaments that can be identified in a specified area has an assumed relationship to ground water productivity. Previous studies of lineament spacing based on geophysical data are reported e.g. in Henkel (1979) for magnetically indicated fracture and fault zones, and in Henkel (1992) for VLF (Very Low Frequency) indicated low conductivity zones, or for topographic escarpments (Henkel, 1992; Henkel et al., 2005). VLF data are sensitive to contrasts in electric conductivity in soil and bedrock. In regions with thin soil cover and underlying high resistive
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Fig. 1. Lineaments and displacements. Steeply inclined faults (like normal faults) result in straight traces (in geophysical or morphological data) enclosing angular blocks. Inclined faults (thrusts) result in curved lines. Steeply inclined shear zones may produce anastomosing traces enclosing shear lenses.
crystalline bedrock, conductive features such as steeply inclined fracture zones in the underlying bedrock can be mapped with VLF measurements (Oskooi, 2004), and are commonly approximated by straight lines (Henkel, 1992). Lineaments and fractures usually have complex azimuth distributions, as in the example given in (Fig. 2a and b), which is based on the interpretation of magnetic anomalies (Henkel, 1979). We also found in the analysis of VLF data that three to four clearly separable azimuth sets can be identified in one area. A case as illustrated in (Fig. 2b) requires an advanced tool for the separation of azimuth sets. Lineament studies are carried out in three steps, i.e., lineament extraction, derivation of lineament properties and their interpretation in a geological (or other) context. Lineament extraction is a challenging area and several authors have provided different tools. Some have attempted to extract lineaments using computer tools (e.g. Burdick and Speirer, 1980; Raghavan et al., 1995). The aim of this paper is to deal with the second of the three steps. For the derivation of spatial properties of lineaments, several approaches can be made. Two methods are presented in Clark and Wilson (1994). One method allows the user to handle a large number of data sets and is generally applied to satellite images. The sampling area can either be a square or a rectangular grid (Kumar and Reddy, 1991), or circular (Mostafa and Qari, 1995). In the other approach each lineament is considered and the relations to its neighbors are derived. Clark and Wilson (1994) developed software to derive the spatial properties of parallel lineaments for such an approach. It requires parallel lineament sets defined by end coordinates and mid coordinates. A computer program in C language derives the spatial properties length, orientation, and parallel spacing. The lineament density can be expressed in three ways: as the number of lineaments per unit area, as the total length of lineaments per unit area, or as the number of cross points per unit area (Mostafa and Qari, 1995; Kim et al., 2004). The results of those methods that derive new indirect measures of spatial properties are intrinsically difficult to interpret in terms of the expected geological concepts like faults and fractures. The increased coverage with detailed aero-geophysical data and the lack of relevant tools to derive spatial properties which are representative for a study area,
encouraged the design of a software tool to determine the spatial properties of lineaments more systematically. The first version of our program was tested and validated on lineaments derived from low altitude airborne Very Low Frequency (VLF) electromagnetic measurements. Although we use some of our previous modules in Ekneligoda (2004) this version extends the analysis in several ways in order to improve the results and provide an enhanced method to carry out lineament studies. The new version allows the user to select the appropriate azimuth set in an interactive way and provides a facility to analyze the other azimuth sets subsequently. Statistical outliers can either be excluded or, in the case of spatial outliers, be treated as individual sets. The proposed Spatial Analysis of Lineaments (SAL) tool can be used for any set of lines and therefore for a range of various applications. All lineaments are included and treated individually to derive statistical distributions of the spatial properties length, azimuth, spacing and the spatial (2-D) frequency. The characteristic statistical parameters average, mode, and standard deviation are calculated for azimuth, length, and spacing. In the next section the basic definitions applied in the software tool are explained. This is followed by a description of the methods developed for the analysis, with emphasis on the spacing and spatial frequency calculation, and then the different steps of an analysis are described with reference to the display windows where the user can interactively design the analysis.
2. Definitions 2.1. Map area The study area is specified by giving the lower left and upper right corners in planar coordinates. The user can select any type of units, but it is recommended to use SI units. 2.2. Lineament The term lineament is used to cover any type of linear feature in any kind of data. These could be traces of discontinuities in a
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distribution, which can be representatively described by the mean value and the variance. The observed distribution may be skewed or contain a few members with a non-characteristic azimuth. Such azimuth outliers can be analyzed as a separate set. 2.4. Length The lineament length is the distance between the start and end coordinates, scaled to a systematic unit (preferably in m). The length distribution for given azimuth set is in theory exponentially decreasing and would contain values from 0 to infinity. In practice, the lineament length is restricted to the smallest spatial resolution (like the spacing between flight lines) at the low end, and to the diagonal of the study area at the high end. In this case the distribution is best represented by the mode value. The distribution may contain atypical members, length outliers, for which the user has to decide if they should be included or excluded in the analysis. 2.5. Spacing The spacing is the perpendicular distance to the nearest lineament of the same azimuth set, and is scaled to the same unit as length. The spacing between two lineaments is not constant throughout their lengths. Since the lineaments usually are not exactly parallel, the spacing is calculated for lineaments within a restricted azimuth range. The spacing distribution depends on the configuration of the search window which the user has to define. With a large search window, the spacing distribution tends to be bi-modal with an increase of the number of multiples of the first mode. This issue is further discussed in the next section. 2.6. Lineament occupancy area A user-defined circumscribing polygon, that covers the lineament set studied, is the lineament occupancy area. The rules for the definition of the construction of the circumscribing polygon are given in the methods section. It is quite common that some lineaments of an azimuth set are located outside their main spatial cluster as spatial outliers (Fig. 3). In connection with the spacing calculation, such spatial outliers should be excluded. If the lineament frequency for several direction sets within an area is required, a separate determination of the common polygon for all lineament sets has to be made. The spatial frequency of lineaments, (or lineament density) is defined here as the two-dimensional frequency, i.e., the total length of a one or several lineament sets divided by the area it Fig. 2. (a) Example of a traditional lineament interpretation based on aeromagnetic data in a 250 500 km region in Northern Sweden (from Henkel, 1979, numbers and letters along frame denote national map sheet identifications). (b) Corresponding azimuth distribution. (Vertical axis shows frequency in %, and horizontal the azimuth in degrees, respectively.)
rock mass or in the earth’s crust. Lineaments are approximated as straight lines and are represented by their planar start and end coordinates, respectively, as seen in the left part of (Figs. 1 and 2). 2.3. Azimuth The azimuth is defined as the angle between a lineament and the grid north and can vary within a 1801 interval. The azimuth distribution of lineaments in a given map area may contain several azimuth sets of lineaments with similar azimuth as in (Fig. 2b). The azimuth distribution for a single set is considered to be a normal
Fig. 3. Examples of spatial outliers (dark gray and outside polygon encircling main spatial cluster).
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occupies. The unit is thus [length]/[length]2, i.e., m number of lineaments per unit area.
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and not the
3. Methods 3.1. Visualization methods The interactive SAL software is based on display windows where the user can set display parameters and enter decisions about parameters that control the subsequent steps in the analysis. Updates of diagrams due to changed visualization parameters are linked and presented automatically. In the displayed diagrams, map frames and axes are displayed in black, lineaments and histograms in red, and added information (like polygons or mode values) in blue. The results from all calculations that concern spatial properties are displayed as histograms, together with the information about the statistical properties average, mode, standard deviation, and number of lineaments. Special attention was therefore given to improve the design of histograms. The frequency scale of histograms can be designed by the user, who can choose between the ranges 20, 50, or 100%. A variable minor subdivision, coupled to the major, is also provided to read off the intermediate values. The distribution of azimuths is usually symmetric and the mean is therefore considered to be representative. The distribution of lengths is however commonly positively skewed and preferably represented by the mode value. The distribution of spacing has a tendency to become bimodal when large search windows are used. In that case the first mode is selected as the mode value. The mode value is calculated as the mid value of the class that has the highest frequency. It is thus dependant on the selected class width. To illustrate the azimuth distribution, three different class widths are provided viz. 21, 51, and 101. The distribution is illustrated both in Cartesian coordinates as a histogram and in polar coordinates as a rose diagram (Henkel, 1968). These two ways of visualising are linked so that the rose diagram is updated when the histogram settings are changed. The range of the length and spacing histograms can be set individually to three different values: 1000, 2000, or 5000 units. This provides a better resolution adapted to the actual ranges of length and spacing.
3.2. Lineament orientation This procedure starts by checking the lineaments for their orientation and switches, when necessary, the start and end coordinates into the required order.
3.3. Lineament length adjustment For measurements that are arranged in a rectangular grid, the length of lineaments is reduced with respect to the measurement spacing and the azimuth of the lineament set, (Fig. 4). This modification is made for individual azimuth sets and the user has to determine an appropriate spatial resolution number.
3.4. Azimuth re-assembling In cases where azimuths occur outside the interval 0–1801, a routine is built to re-arrange the distribution into a continuous histogram for the interval 0–1801.
Fig. 4. Interpreted length of lineaments is reduced with spatial resolution, which may be different in different directions, i.e., depending on angle v to measurement grid. (Figure shows situation for airborne measurements where measurement spacing is smaller along flight lines and not necessarily aligned to a straight line.)
3.5. Separation of azimuth sets If several sets of azimuths are encountered, the selection of one azimuth set is required to continue the analysis. The user can define the lower and upper values of the azimuth angle based on the frequency minima occurring in the azimuth distribution. The accuracy of determining the lower and upper value can be enhanced by selecting the smallest class width (Fig. 5).
3.6. Calculation of spacing The spacing between two lineaments is calculated by creating roughly equi-dimensional blocks between the two lineaments. A typical variation of spacing is illustrated in Fig. 1 in the paper by Ekneligoda (2004). The program first calculates the perpendicular distance (d1) between the current lineament and its neighbor, then it goes forward the distance d1 along the current lineament, and draws the second perpendicular projection to the neighbor line (d2). Next, it moves the distance d2 along the current lineament to draw the third perpendicular (d3) projection. This procedure continues until the lineament is not long enough to match the extent of (dn). The selection of several distance calculations between pairs of lineaments gives a better spatial representation of the spacing distribution, as the calculations are spread over the entire area where the set of lineaments occurs. How spacing is calculated, illustrated for a strictly parallel set of lines (in a real case all directions within one azimuth set are considered), is presented in (Fig. 6). The current lineament is shown with the thicker line with a rectangular node. The same set of lineaments is shown with three examples of spacings calculated (or refused) related to a chosen current lineament. The non-resulting test for a subsequent spacing is only illustrated for the upper left case in (Fig. 6). The search window is the area on each side of a lineament within which the distance to a neighbor lineament is calculated, as illustrated in (Fig. 6). To include all possible spacings between neighbor lineaments, a front window and a back window is created, based on the length distribution of a set of lineaments. As lineaments can occur in different lengths and with different orientations, the use of both windows is required, as otherwise there is a risk of losing candidate neighbor lineaments. SAL provides the facility to vary the size of the search window in the off lineament (perpendicular) direction by multiplying the
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Fig. 5. Example of separation of azimuth sets by using a smaller class width. (Vertical axis shows frequency in %, and horizontal the azimuth in degrees, respectively.)
Fig. 6. How spacing is calculated is illustrated for strictly parallel set of lines (in a real case all directions within one azimuth set are considered). Same set of lineaments is shown with 3 examples of spacings calculated (or refused) related to a chosen selected lineament. Non-resulting test for a subsequent spacing is only illustrated for upper left case. User-defined size of calculation window is critical as it defines which of the neighbor lineaments are considered for a spacing calculation.
mode of the length distribution with a factor 1, 1.5, or 2 to find the nearest lineament. Different window sizes can be tested to find the optimal coefficient. For this purpose, a map is provided
containing the lineament arrangement and the first calculated spacing (i.e., d1 in (Fig. 1) in Ekneligoda and Henkel (2006)) for each pair of lineaments.
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3.7. The spatial occupancy polygon The polygon is drawn interactively by joining the endpoints of the outer lineaments of a spatial cluster as illustrated in (Fig. 3). The user-defined polygon is used to remove the spatial outliers and for the definition of the area used for the spatial frequency calculation. The user can make any complex shaped polygon to exclude empty areas. Lineaments that are cut by the polygon are included in the subsequent calculation. The user-defined polygon is expanded to get a more representative area. This concept is connected to the common occurrence of gaps in the spatial distribution of lineaments. As a provisional approach, the expansion is related to both the length and spacing statistical properties. In the lineament direction the expansion is based on the length statistical properties, whereas in the perpendicular direction the expansion is based on the spacing statistical properties. This complex expansion criterion is simplified by taking the average value of both length and spacing mode values. The expanded polygon is displayed in (Fig. 10) together with the result of the frequency calculation.
4. Spatial analysis of lineaments (SAL)
Fig. 7. Work path for lineament analysis using SAL tool.
The SAL software was first built using Visual Basic 6, which is considered to be a user-friendly computer language partly due to its advanced graphical interface (Peasley and Simon, 1998; Perry, 1999; Foxall, 2002). The new extended program consists of five display windows and several modules. The work flow chart of the extended program is presented in (Fig. 7).
Fig. 8. Display window 2. Azimuth distributions with several lineament sets, spatial outliers and rose diagram are shown. (Vertical axis shows frequency in %, and horizontal the azimuth in degrees, respectively.)
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Fig. 9. Display window 3. Histogram and statistics of azimuth (left, in degrees) and length (right, in m) of lineaments when spatial outliers have been removed. (Vertical axis displays frequency in %.)
In the first display window, the user is asked for the location of the database, which contains the starting and ending coordinates of the lineaments. The user can move to the second display window by clicking the button provided in the lower right portion of the window. In this display window the azimuth is shown in the histogram form and as a rose diagram and a map view of all lineaments is presented in (Fig. 8). In this window the polygon for the lineament density determination has to be defined. Once the azimuth set is selected, a map view of its lineaments is provided next to the azimuth distribution. At this point, the user can move to the third display window, where the compilation of length and azimuth is presented in (Fig. 9) for the selected set after excluding spatial outliers. Two histograms are provided for the interaction with the user. The scales of the histograms can be adjusted according to the range of each property azimuth, length, and the frequency, respectively. The histogram parameters’ range and class width are also displayed. The save button can be used to save the statistical values of length and azimuth. The spatial frequency (or lineament density) calculation is made in the same window. This procedure requires the definition of the area that is occupied by the lineaments that is studied. In most situations this area has an irregular outline, as seen in (Fig. 3). The user has to define this area as the polygon that connects the outer ends of the lineaments. Then the user can move to the fourth display window (Fig. 10) where the spatial arrangement of the lineaments is illustrated. The first spacing calculations, for each lineament are shown with different colours for back and front window. The list box shows the coordinates between which the spacings were calculated, and the corresponding spacing values. As the expansion length of the polygon is based on the average value of length mode and the spacing mode, the facility has been provided to add the length
histogram on the spacing histogram at this level. Each mode value together with their average value is also drawn with the corresponding color. The save button can be used to save the statistical values of spacing and lineament frequency. In the final display window (Fig. 11), the histograms and the statistical properties: mean, standard deviation, mode, and number of lineaments are summarized for each spatial property. In addition to azimuth, length and spacing histograms a rose diagram of the azimuth distribution and the user-defined polygon for lineament density calculation is also presented. After analyzing an azimuth set, the user can go to the next set by using the ‘Next azimuth set’ command. This procedure is repeated for all the azimuth sets without closing the program. The new program allows the user to handle about 30–40 azimuth sets though in reality maximum number of azimuth sets may be less than 10. The save button can be used to save all statistical values with all spatial properties.
5. Discussion The spatial precision of the entered lineament coordinates depends on the spatial resolution of the original data from which the lineaments have been derived. The pointing uncertainty in connection with the on-screen digitizing is accounted for by reducing the lineament length with the spatial resolution. Airborne measurements in particular have a strongly anisotropic sampling pattern, which may require a more complex approach of the spatial resolution, as it is different in different directions and hence affects the lineament length to a variable degree. Therefore it is recommended that the user has to find a suitable reduction measure to each individual direction set of lineaments.
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Fig. 10. Display window 4. Histogram of spacing and length (in m) and a map of first set of spacings in gray tones (in software green within front window, red within back window) together with lineaments considered in analysis. It also displays user-defined polygon and expanded polygon. (Vertical axis displays frequency in %.)
The common situation is that all lineaments appear together in one map. In some cases a separation into different azimuth sets is not obvious as lineaments azimuth angles can differ by few degrees only. It is always advisable to use a histogram of the azimuth for the separation of different azimuth sets. In our new tool we have provided an enhanced facility in order to carry out an intelligent azimuth separation. Some lineaments can appear away from the main cluster (spatial outliers). These outliers may not contribute to any spacing values in the subsequent stages as they are too far away from their neighbors. In such a case it is appropriate to remove them from the set before the subsequent statistical analysis is made. Our tool provides a method to remove these spatial outliers interactively. The user can identify the outliers at the beginning as the map view is provided next to the azimuth distribution. After drawing a polygon in an appropriate way the user can take away the outliers just by using the outlier exclusion button. This way the representativity of the statistical properties of azimuth, length, spacing, and lineament density has been increased. If the user is interested in getting the information about azimuth and length outliers, this information can be collected by selecting the corresponding azimuth interval. In the design of diagrams showing results, the user has several choices to find a suitable scaling of the histograms. The calculation of the mode value as the mid value of the class with the highest frequency is valid, when the right and left side frequencies are the same. In other cases a negligible error is admitted. The spacing calculation can be carried out by using different sizes of the search window. The user can choose an appropriate
factor. The use of a higher factor adds longer spacing values by connecting more lineaments thus contributing the right tail of the spacing histogram. This may lead to bi-modal spacing distribution. The use of unit factor on the other hand significantly cuts off the higher spacing values and reduces the cluster of lineaments for the subsequent analysis. The statistical representativity of a derived property for an area depends on a representative sampling scheme. An (at least roughly) equi-dimensionally spaced sampling should thus be attempted. That is the reason to design the sampling of spacing values at distances that are comparable to the spacing itself. The lineament density is derived based on a circumscribing polygon defined by the user (Fig. 3). This polygon should not include empty areas and be drawn close to the end of lineaments. Lineaments that are cut by the polygon will be included in the calculations. Considering the uncertainty of the gap length, the polygon construction does not require a painstaking precision (just round the lineaments) and if required, a test with differently drawn polygons may reveal the influence on the lineament frequency. The representative area of a set of lineaments reaches outside the lineaments. In the direction perpendicular to the strike this would correspond to one half spacing distance. In the direction parallel to the lineaments it would correspond to half the length of the gaps between lineaments, i.e. the distance between perpendicular lines at the start and end of neighboring lineaments. As the distribution of the gap length may be similar to the distribution of length, we have approximated the expansion length in the strike direction with the mode of the length
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Fig. 11. Summary of results uni-directional azimuth (histogram and rose diagram), length and spacing distribution together with map view of lineaments and polygons involved in lineament frequency calculation are presented. Also, unidirectional lineament density is shown.
distribution. To expand the polygon to the representative area, a compromise solution is given by using the average of the mode value of the length and the spacing distribution. We believe our new compromise solution results in a more representative area as it takes into account both length and spacing simultaneously. In case the area is a window out of a larger continuously represented region, the area of the window itself should be used for the calculation of the spatial frequency. If the studied region contains several spatial clusters of lineaments, it is advisable to subdivide the lineaments into several spatial sets in order to comply with the requirements for representativity of derived properties. A similar reasoning can be made regarding empty areas and a suitable subdivision may reduce the under-sampling effect that otherwise would be caused. Another factor of importance is the dimension range within which the sampling is made. Lineaments that represent fracture lengths are usually part of a fractal continuity ranging over many orders of magnitude (Gillespie et al., 1993) The observed dimension range is restricted by the area dimensions (the spatial coverage) of the surveyed region on the high end, and by the spatial resolution of the measurements on the low end. The distribution of fracture length gives an indication of the presence of one or several categories with different characteristic lengths. In a further development of the spatial analysis of lineaments, statistics about the gap length can be calculated and used for a more precise determination of the occupancy area. The multiple fracture frequency, based on a common polygon for all azimuth sets is under construction. In the case of shear zone geometry of lineaments, additional modules can be designed to describe other
relevant spatial properties like shear lens shape and shear lens dimensions.
6. Conclusions The presented tool for spatial analysis of lineaments (SAL), is a versatile interactive software tool for detailed characterization of lineaments based on the statistical criteria. The tool can handle a large amount of any line features at any scale and presents the results of an analysis in quantitative properties that are spatially and statistically representative. The spatial properties of different azimuth sets are derived in sequence, thus avoiding the repetitive calling and opening of the original database. Statistical outliers of the spatial properties can be removed prior to the calculations which improve the representativity of derived results. If spatial outliers cluster in some part of the study area, they can be analyzed as a separate group by a mutually excluding procedure. Complex azimuth distributions can be separated interactively into individual azimuth sets in a logical way with the zooming function by which the class width can be reduced. SAL provides a compromise solution for the determination of the polygon for the lineament density calculation by taking the average value of the length and spacing mode as the expansion length. A summary of results is provided in graphical form for presentation and comparison purposes. Our software would be useful in building a fully automated interactive analysis tool starting from the lineament extraction level. The software shows a path to also analyze shear
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zone related features, provided the spatial description of shear zones can be given in planar coordinates.
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