Point feature class

Normal fault traces (polylineshapefiles) were converted to their midpoints (point shapefiles)using the Feature to Point (Data Management) tool.

3.2.4.1 Multi-Distance Spatial Cluster Analysis

The Multi-Distance Spatial Cluster Analysis tool, based on Ripley's K-function (Ripley, 1977), was used in the Spatial Statistics package of ArcGIS 10 to determine the distance to which the BR and CF systems display a clustered pattern for each caldera. The tool reveals how an observed spatial pattern (clustered or dispersed) of a variable (fault trend) changes over dif- ferent spatial scales (distances) (e.g., Dai et al., 2012). Compared to other spatial analysis pat- tern identification methods (e.g., autocorrelation) that calculate the distance between neighboring values of a variable, the K-function gives a measure of the spatial pattern over a range of dis- tances. In this method, the number of distances to evaluate, and the distance and/or distance in- crement, are specified, and used to compute the average number of neighboring features that are within the specified distance (e.g., Wong and Lee, 2005; Scott and Janikas, 2010; Dai et al., 2010; Streib and Davis, 2011).

In this study, the location of each fault trace is given by the midpoint of its trace. The method incrementally draws circles (buffers) around the midpoint of an individual fault trace (si, target variable) at specific radial distances ( lags, d). It then calculates the number of points located within each circle (i.e., for each lag), and moves to the next target point (sj), and repeats

the process for all points (fault trace midpoints). The number of distance bands (lags) is specified in the GIS application. If two points with the greatest distance are located at a distance d from each other, then, the number of the lags to hold the points is equal to d/l, where l is the distance increment (l < d). The iteration number ranges between 1 and g=d/l. The distance bands which are developed around each point (i) are called K. The value of the K statistic at a given distance (d) is given by:

(Eqn. 3.3)

where |A| is the area of the study area, n is the number of the fault trace midpoints, and i and j are the indices of the target fault trace and other fault trace midpoints, respectively. The || si-sj ||

defines the distance l between si and sj, and is the average density of the fault trace mid- points placed in each circular buffer (with a size of l x g) divided by the average density of all midpoints in the study area. If the average concentration of fault trace midpoints in a given circular buffer is greater than the average density of fault trace midpoints in the whole area, the distribution of fault trace midpoints is considered to be clustered for that distance (l). The distance between a target point (i) and other points (j) is multiplied by the weight factor I(dij). The binary I(dij) is 1 if the neighboring fault trace midpoint (j) is within the distance band of the target fault trace midpoint (i), and is 0 if no neighboring fault trace midpoint falls within the distance band of the target fault trace midpoint.

ArcGIS calculates an expected K(d) value for a random distribution applying a user- defined number of permutations (9, 99, or 999) under the Compute_Confidence_Envelope option. The optional values lead to the random placement of 9, 99, or 999 sets of points for analysis. If the no weight field is specified, the confidence envelope is built by randomly

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selecting points and calculating the K(d) value for them. For example, if the user selects 9 permutations, the tool randomly selects (distributes) 9 points, for each iteration, and determines the K(d) value for each distance that deviates from the Expected K(d) value for random

distribution by the greatest amount. These values define the confidence interval (Esri, 2012). For a random point pattern, a theoretical assessment of the K-function is πl2, where l is the lag distance. In comparison, the observed K-function is less than πl2for a dispersed pattern, and is greater than πl2 for a clustered pattern.

Figure 3. 3. Ripley’s K-function calculation tool showing the distance bands (buffers) plotted around eve-

ry point (O’Sullivan and Unwin, 2010).

The results of the K-function analysis is commonly interpreted by plotting the K(d) val- ues (y-axis) against distance (x-axis) on a graph. Because the amount of K(d) values increases with increasing distance, these values need to be adjusted by converting them into a normalized measure of the K-function by the square root transformation of K(d), which is known as L(d) (Barot et al., 1999).

L(d) represents the difference between the value of the observed K-function and that of the K-function for the expected random pattern. L(d) is calculated from the following equation:

(Eqn. 3.4)

where L(d) is the value of L at a given distance (d), A is the area containing the features, k(i,j) is a weight, and n is the number of points (fault trace midpoints). Each distance band will be mul- tiplied by the weight k(i.j) which is 1 if the distance (d) is less than the distance between two points i, j (no edge correction), and is zero if it is not. The sum of distances is then multiplied by the area of the study are, and the result is divided by π times the number of points less one (i.e.,

the remaining points other than the selected one). The square root gives the value of L at a given distance (d).

Comparing the observed K(d) value with the expected K(d) value allows us to examine the spatial pattern (clustered or dispersed) of the points. The expected value, in any given distance d, gives a random distribution which is a line at 45o on the L(d) versus d plot (Figure 3.4). At a given distance or scale of analysis, the points show a more clustered distribution pattern than a random distribution if the observed L value line is above both the expected

(random) L value and the higher confidence envelope lines. The points display a more dispersed pattern, compared to the random distribution, when the observed L is lower than the lower confidence envelope, and both plot under the expected L value line (Bailey and Gatrell, 1995; Boots and Getis, 1988; Anselin, 2003; Morrison et al., 2004; Mitchell, 2005; Smith et al., 2007).

The size and shape of the study area influence the results of the K-function (Mitchell, 2005) because points located near the edges of the map area are likely to have less neighboring points in the buffer zone outside of the area. This problem can be fixed applying two correction

methods: (i) the Simulate Outer Boundary Values edge, in which the fault trace midpoints across the analysis area boundary are mirrored to correct for the underestimated values near the

boundaries, (ii) the Reduce Analysis Area edge correction method that reduces the size of the study area to the distance which is equal to the largest distance band to be used in the analysis (Cressie, 1991; Yamada and Rogerson, 2003; Mitchell, 2005).

Figure 3.4. Plot of L(d) against d for a truly random distribution (straight line). L(d) measures the differ- ence between the observed pattern and that expected under the CSR model.

In this study, the distance increment or lag l was selected to be equal to 10, the

orientation (azimuth) of fault traces was specified as weight, and the simulate outer boundary values were chosen to fix the edge effect problem. The weighted K-function tool was run with 99 iterations for complete spatial randomness in which 99 sets of polygon centroids were randomly placed to calculate and plot the K(d) value and the confidence envelopes above and below the Expected K(d).

3.2.4.2 Directional distribution using Standard Deviational Ellipse (SDE)

The Directional Distribution Tool, in the Spatial Statistics package of ArcGIS 10, was used to determine the standard deviational ellipse, representing the trend of the geographic dis- tribution (dispersion) (Scott and Janikas, 2010) of the midpoints of the traces of the Basin and Range faults in spatial domains I and II, and the cross faults in the three spatial domains of each of the five temporal domains described in Chapter 2.

The tool reveals the principal direction of the maximum distribution of the midpoints

from a set of polylines (fault traces) in each spatial domain, by displaying a directional bias for the set (Wong and Lee, 2005; Allen, 2009; Bigham and Sanghyeok, 2012). Each standard devia-

tional ellipse is described by its major and minor axes and the azimuth () of the major axis measured clockwise from the North. This method first calculates the coordinates of the mean center or spatial mean (i.e., central or average location) of the midpoints of the set of fault traces in each domain based on the following formula:

(Eqn. 3.5) (Eqn.3.6)

where xi and yi are the coordinates for midpoint of the fault trace i, {x , } are the coordinates of

the mean center for the fault traces, and n is the total number of fault traces. The angle of rota- tion (), representing the azimuth of the long axis measured clockwise from North is calculated as follows:

The method calculates the standard deviation of each of the midpoints relative to the mean center, and displays the maximum and minimum spatial dispersion of the points, around the mean, with an elongated circle, or an ellipse, with the major axis oriented to match the spatial dispersion of the midpoints of fault traces (Wills et al., 2011). The standard deviations for the x-

axis and y-axis (x and y) are estimated using the following formula, where and are the

deviations of x- and y-coordinates from the mean center:

(Eqn. 3.8)

(Eqn. 3.9)