Analysis of a Spatial Point Pattern: Examining 

the Damage to Pavement and Pipes in Santa 

Clara Valley Resulting from the Loma Prieta 

Earthquake 

By G.A. Phelps 

Open File Report 2007–1442 

2008 

U.S. Department of the Interior 

U.S. Geological Survey 
U.S. Department of the Interior 

Dirk Kempthorne, Secretary 

U.S. Geological Survey 

Mark D. Myers, Director 

U.S. Geological Survey, Reston, Virginia 2007 

For product and ordering information: 

World Wide Web: http://www.usgs.gov/pubprod

Telephone: 1-888-ASK-USGS 

For more information on the USGS—the Federal source for science about the Earth, 

its natural and living resources, natural hazards, and the environment: 

World Wide Web: http://www.usgs.gov

Telephone: 1-888-ASK-USGS 

Suggested citation: 

Phelps, G.A., 2007, Analysis of a spatial point pattern: examining the damage to 

pavement and pipes in Santa Clara Valley resulting from the Loma Prieta earthquake: 

U.S. Geological Survey Open File Report 2007-1442, 49 p. 

[http://pubs.usgs.gov/of/2007/1442/]. 

Any use of trade, product, or firm names is for descriptive purposes only and does 

not imply endorsement by the U.S. Government. Although this report is in the public 

domain, permission must be secured from the individual copyright owners to 

reproduce any copyrighted material contained within this report. 

2
ABSTRACT 

This report describes some simple spatial statistical methods to explore the 

relationships of scattered points to geologic or other features, represented by points, lines, 

or areas. It also describes statistical methods to search for linear trends and clustered 

patterns within the scattered point data. Scattered points are often contained within 

irregularly shaped study areas, necessitating the use of methods largely unexplored in the 

point pattern literature. The methods take advantage of the power of modern GIS toolkits 

to numerically approximate the null hypothesis of randomly located data within an 

irregular study area. Observed distributions can then be compared with the null 

distribution of a set of randomly located points. The methods are non-parametric and are 

applicable to irregularly shaped study areas. Patterns within the point data are examined 

by comparing the distribution of the orientation of the set of vectors defined by each pair 

of points within the data with the equivalent distribution for a random set of points within 

the study area. A simple model is proposed to describe linear or clustered structure 

within scattered data. 

A scattered data set of damage to pavement and pipes, recorded after the 1989 

Loma Prieta earthquake, is used as an example to demonstrate the analytical techniques. 

The damage is found to be preferentially located nearer a set of mapped lineaments than 

randomly scattered damage, suggesting range-front faulting along the base of the Santa 

Cruz Mountains is related to both the earthquake damage and the mapped lineaments. 

The damage also exhibit two non-random patterns: a single cluster of damage centered in 

the town of Los Gatos, California, and a linear alignment of damage along the range front 

of the Santa Cruz Mountains, California. The linear alignment of damage is strongest 

between 45o and 50o northwest. This agrees well with the mean trend of the mapped 

lineaments, measured as 49o northwest. 

 

INTRODUCTION 

Scattered point data are common in the geological sciences; earthquake 

epicenters, the locations of mineral deposits, and oil plays are examples of geologic data 

that are represented as point data. Two questions are common with these data: “Are the 

points related to some other geologic feature?” and “Do the points themselves exhibit a 

pattern?” 

One of the difficulties of statistically analyzing scattered datasets is that the data 

often occur within irregular study areas. A null hypothesis of randomness, in this case a 

set of randomly located points, is often the starting point for any analysis. Complete 

spatial randomness for points is defined as a set of points whose locations are an 

independent random sample taken from a distribution of equal probability across the 

study region (Diggle, 2003). This implies that the location of a given point is 

independent of the location of any other point. In other words, the points do not interact 

with one another. 

A typical method of generating distribution parameters of point data sets 

satisfying complete spatial randomness within an irregular study area is Monte Carlo 

simulation, where randomly located points are added to a study area and the properties 

are averaged over 100 or more simulations (Diggle, 2003). While this method is robust, 

it often requires programming effort to meet the needs of the particular analysis, requires 

computing power, and does not necessarily offer any insight into the problem. 

3
Some distributions can also be estimated more directly than by the use of Monte 

Carlo simulations. Such direct methods commonly require less computing time, less 

programming effort, and ultimately offer more insight into the problem at hand. As will 

be seen, for example, distributions based on a null hypothesis of complete spatial 

randomness can be conveniently estimated in at least two cases by using powerful raster 

processing tools available in modern GIS software packages. 

The first case, the problem of investigating the spatial relationship of scattered 

point data to other objects, was described by Okabe and Fujii (1984). The second case, 

the problem of investigating the randomness of a scattered dataset itself, can be derived 

from principles developed in solving the first case. The following sections are devoted to 

exploring methods of approaching both problems, using a dataset of damage to pavement 

and pipe breaks caused by the 1989 Loma Prieta earthquake as the point dataset, and a set 

of mapped areal photographic lineaments as the objects which may be spatially related to 

the earthquake damage. 

DATA AND SOFTWARE 

The 1989 Mw 6.9 Loma Prieta earthquake (U.S. Geological Survey, 2007), the 

epicenter for which was located roughly 100 kilometers south of San Francisco, 

California, caused damage to many types of public works, including roads and sidewalks, 

throughout Santa Clara Valley (figure 1). The location and type of damage were mapped 

via extensive fieldwork and database compilation (Schmidt and others, 1995). 1427 

observations, taken from fieldwork, the records of utility and transportation institutions, 

and local governments, were recorded for a 663 square kilometer area. Schmidt and 

others (1995) exhaustively searched roads and parking lots within the study area, 

measuring contractional damage indicators for ground level damage, pavement breaks 

and curb breaks. They added to this field data by incorporating damage for sub-surface 

gas and water line ruptures, data shared by local utility, transportation, and governments. 

In all, five types of damage were recorded (asphalt, channel lining, concrete, gas line, 

water line), in addition to the sense of deformation (if any), the freshness of the damage, 

and whether or not the damage was at or below the ground surface. Damage to structures 

was not included because it depends upon building construction, materials, and design, 

and the damage cannot be constrained to the Loma Prieta event. Sidewalks, pavement, 

and other public infrastructure works are more commonly built to uniform specifications 

and can therefore be used to detect ground motion in a consistent manner. In this report 

only the location of the damage is considered in the analyses; analyses based on subsets 

of the data would be a reasonable next step for further research. For further information 

discussing the 1989 Loma Prieta earthquake the reader is directed to U.S. Geological 

Survey Professional Papers 1550-1553. 

The (unpublished) digital version of damage recorded in Santa Clara Valley is 

used as the example scattered data for analysis in this report. While the data were 

collected by plotting the locations on 1:24,000-scale topographic maps (before the 

widespread use of GPS), subsequent digitizing and registration to scanned topographic 

maps indicates a relatively high level of positional accuracy, where data points often plot 

on the correct side of the street. While not all points were checked, this observation 

suggests the data points are accurately located to within a few meters. 

4
Hitchcock and others (1994) mapped a series of areal photographic lineaments as part of 

a Quaternary geologic mapping study to investigate seismic activity along the eastern 

range front of the Santa Cruz Mountains, where the mountains abut Santa Clara Valley. 

The lineaments are based on several types of geomorphic and photographic features, such 

as topographic scarps and depressions, stream sinuosity, vegetation lineaments, and tonal 

changes in the ground surface. The lineament database used in the next section is an 

unpublished digital rendition of geology from plate 2 of Hitchcock and others (1994). 

The GIS software used to test and implement the methods described in this report 

was the commercial package Arc/Info, and the associated product ArcMap (ESRITM). 

Arc/Info has built-in tools to generate many of the analytical results used in this report, 

including density maps, buffers, nearest-neighbor distances, point-in-poly operations, and 

database merges and queries. Appendix II contains pseudo-code for the analytical 

operations performed in this report. 

The commercial statistical software package Splus (InsightfulTM) was used for the 

statistical analysis, including graphical displays, analysis of distributions, and goodnessof-fit tests. 

5
METHODS AND RESULTS 

In order to compare the observed pattern of a scattered point dataset to a null 

hypothesis of a pattern of complete spatial randomness one must be able to describe the 

null distribution for the study area in question. This is often done in the literature by 

means of quadrat analysis, where the study area is broken up into squares of equal size 

(quadrats) and the frequency of the points per quadrat generates a Poisson distribution 

under complete spatial randomness (Diggle, 2003; Upton and Fingleton, 1985). One is 

not always fortunate enough to have a rectangular study area, however. The natural 

world often precludes rectangular study areas by imposing natural boundaries (e.g. rivers 

and lakes, forests, cliffs and other steep terrain), and existing datasets were not always 

created to conform to a rectangular study area. If one does not wish to be limited in 

analyzing scattered datasets, methods must be developed to describe the distributions of 

complete spatial randomness for irregular study areas. 

The question “Are the points in a given scattered dataset related to some geologic 

phenomenon?” can be addressed by considering the geologic phenomenon as fixed, and 

the points as realizations of some process that is perhaps influenced by the phenomenon. 

Points are considered “events” that occur with some probability within the study area. 

For the null hypothesis of complete spatial randomness, the study area is considered 

homogenous; that is, a point has equal probability of being located anywhere within the 

study area, independent of features (e.g. topography, soil type, vegetation). Now suppose 

that this assumption is violated: some geologic phenomenon creates inhomogeneities 

within the study area or some geologic process causes the points to locate non-randomly 

within the study area. If the points and the geologic phenomenon are related, one might 

expect either a positive or negative spatial association between them, in which the points 

would either have a tendency to be attracted to, or avoid, the phenomenon. In other 

words, in the end-member cases the points will either be co-located with the phenomenon 

or as far from it as allowable given space constraints. The strategy, then, is to find a null 

distribution for the distance from the geologic phenomenon, under the assumption of 

complete spatial randomness, within the given study area, and compare this with the 

observed distribution. If the observed distribution of distance from the geologic 

phenomenon is different from the null distribution, then one can determine if two are 

positively or negatively correlated. 

The question “Are the points themselves randomly located within the study area?” 

amounts to determining whether or not the scattered point dataset can be distinguished 

from complete spatial randomness. A new method of examining whether a scattered 

dataset deviates from complete spatial randomness is described. This method relies on 

the orientation specified by pairs of points within the dataset. 

Complete spatial randomness depends on the sample points being independent of 

one another. No point may have an effect on the location of another point. Stated 

differently, if there are two processes influencing the location of points, then the effect 

from one may influence the statistical results calculated for another, and this would 

violate the assumption of independence (of course it would be very unusual to have only 

one process operating in any natural system, but one would like the dominant process to 

overshadow the rest). Before any analysis begins the dataset should be examined for 

obvious effects from secondary processes. 

6
Association of damage points with areal photographic lineaments: an example from the 

Loma Prieta earthquake 

“Are the points in a given scattered dataset related to some geologic 

phenomenon?” is addressed for the example of comparing damage incurred during the 

1989 Loma Prieta earthquake in Santa Clara Valley, CA (figure 2), 

to a set of areal photographic lineaments mapped within an irregular study area along the 

range front of the Santa Cruz Mountains (Hitchcock and others, 1994), near the towns of 

Los Gatos and Saratoga (figure 3). The study boundary is determined by using the full 

extent of the available geologic mapping presented by Hitchcock and others (1994), a 

somewhat irregular strip of mapping along the range front of the Santa Cruz Mountains. 

The example demonstrates the method for determining a spatial association 

between linear features and a scattered point dataset, but the method is easily extendable 

to point features or area features (Okabe and Fujii, 1984; Okabe and others, 1988). For 

convenience, references to damage in the rest of this report will mean the damage from 

Loma Prieta earthquake in Santa Clara Valley as describe by Schmidt and others (1995), 

and the study area covered by the damage data will be referred to as the “damage study 

area.” Similarly, references to lineaments will mean the lineaments mapped by 

Hitchcock and others (1994) and the study area encompassing the lineaments will be 

7
referred to as the “lineament study area.” The lineament study area is completely 

contained within the damage study area (figure 3). 

The lineament data have been cited as evidence to support the interpretation and 

mapping of reverse faults along and outboard of the range front of the Santa Cruz 

Mountains (Hitchcock and others, 1994; Hitchcock and Kelson, 1999). This 

interpretation is supported by geophysical evidence of a steep-sided subsurface basin 

adjacent, and parallel to, the range front (Langenheim and others, 1997). However, while 

some of the lineaments mapped are based on observed geologic features such as 

topographic scarps, others are based on vegetation or tonal changes seen in areal 

photographs. The latter types of evidence are not necessarily indicative of fault activity. 

A spatial association of the damage with the lineaments would strongly suggest the 

features that define the lineaments are related fault activity. If the damage is, on average, 

closer to the lineaments than would be the case for complete spatial randomness, then the 

lineaments and the damage tend to be co-located and are thus spatially related. 

To answer the question of whether or not the damage is associated with the 

lineaments a model of randomness must be constructed with which to compare the 

observed data. Such a model can be developed by using the distance of each damage 

point to the nearest lineament. If the damage and the lineaments tend to be co-located, 

the damage ought to occur on or near the lineaments. If they avoid co-locating, the 

8
damage ought to be as far as possible from the lineaments. A spatially random 

distribution of damage locations would show no association with lineaments throughout 

the study area. 

The distribution of the distance of the damage to the lineaments can be obtained 

by measuring the distance of each damage point to the nearest lineament (considering 

only the subset of the damage data that falls within the lineament study area (as defined 

by Hitchcock and others, 1994)). The problem then becomes how to generate the 

probability density function of the distance to the lineaments for a random process. Once 

this is accomplished the two distributions can be compared.

Assume the lineaments are fixed; that is, they are mapped correctly and properly 

located. This is not an insignificant assumption, but here the mapped features are taken as 

given. Then, over the lineament study area, one can compare the distributions of the 

distance of damage points to the nearest lineaments with the distance expected for a 

randomly distributed set of points. 

To obtain a random distribution, one need only consider the probability of a point 

falling in a particular sub-region of a given study area (Okabe and Fujii, 1984). Since a 

random point has equal probability of falling at any particular place in the study area, the 

probability that it will fall in a given sub-region is simply the area of the sub-region 

divided by the total area (figure 4). 

 

Now consider the sub-region of the study area defined as any point greater than x

and less than x+h distance from the nearest lineament (figure 5). The probability of a 

9
random point being between x and x + h distance from the nearest lineament in the 

lineament study area is simply the area defined by the buffer around the lineaments from 

x to x +h divided by total area. If the probabilities for many buffers (for example, in 200 

m increments) are combined for a sequence of distances from zero to the farthest distance 

in the study area, the result is an approximation of the probability density function for the 

distance to the nearest lineament for a random set of points. 

This distribution can be numerically approximated by rasterizing the lineament 

study area and keeping track of all cells a given distance from the nearest lineament 

(figure 6). The ratio of the area of the sub-regions between x and x + h distance from the 

nearest lineament to the total study area is simply the number of pixels between x and x 

+ h distance from the nearest lineament divided by the total number of pixels in the study 

area (the units cancel). This numerical approximation of the random distribution can be 

compared to the observed distribution of the distance of the damage points to the nearest 

lineament. Since the resulting probability density functions are non-gaussian, a Smirnov 

test (Rock, 1988) can be used to test whether or not the two distributions are different. 

The above method is appropriate provided one assumption is satisfied: that it is 

assumed the observed process is, or at least could be, homogeneous over the study area. 

That is, the process is the same at any location in the study area. For example, as a first 

approximation, damage is assumed to be equally likely regardless of the type of soil that 

10
is present. If the process is homogeneous (does not depend on soil type), it is then 

reasonable to compare this observed distribution with a random distribution that meets 

the same assumption of homogeneity. A study area that includes sub-areas that affect the 

process differently is called inhomogeneous. 

The mapped damage is an example of an inhomogeneous point pattern, because 

the damage data consists of pavement breaks (these occur exclusively on paved areas i.e.: 

streets and sidewalks) and pipe breaks (also almost always occurring beneath streets, 

because utilities tend to follow public streets and rights-of-way). Practically speaking, 

this damage cannot occur outside of an area covered by a street. The process is therefore 

inhomogeneous over the both damage and lineament study area. The total sample space 

is the subset of the study areas that are covered by streets and sidewalks. 

Changing the sample space changes the probability of a random point falling a 

given distance from a line because it changes the total area available to the random point. 

In order to calculate the probability for the inhomogeneous point process the area of the 

streets and sidewalks must be found. 

Finding this subset of the lineament study area would require coupling extensive 

data at the county level for street footprints, if such data could be obtained. A scanned 

1:100,000 USGS topographic map of the streets for the lineament study area gives a 

11
reasonable approximation of the area covered by streets and sidewalks, as can be seen in 

figure 7, 

which compares the scan to the 1995 digital orthophoto quarter-quadrangle for a portion 

of the area. While certainly not error-free, the scan appears to approximate the area to 

within 10% to 20%. In the subsequent analyses the scan will be used to generate the 

subset of the lineament study area for examining the inhomogeneous point process. 

12
Note that the lineaments are assumed to be fixed, and the damage is taken as the 

process that is either related to these fixed features or not. The question could have been 

posed conversely; are the lineaments located more closely to the mapped damage than a 

random process for generating lineaments? In this case the damage points are the more 

stable features. Based on geomorphologic evidence, the lineaments represent features 

that span a range of ages (Hitchcock and others, 1994). If they are tectonic features, then 

one would expect them to span multiple events. In this case the damage represented by a 

single event would not necessarily be associated with every lineament, but with a subset 

belonging to one or more fault strands experiencing activity in the Loma Prieta event. 

Therefore the question is posed such that the particular event is compared with the 

general tectonic framework. 

 

DECLUSTERING THE DATA 

The method described above assumes that the damage is not significantly affected 

by another process. A density plot1

 of the damage data (figure 8, top) shows the linear 

concentration of damage along the range front of the Santa Cruz Mountains apparent in 

figure 2, and the tight clustering of damage near the town of Los Gatos. Previous authors 

(Hitchcock and Kelson, 1999; Langenheim and others, 1997; Schmidt and others, 1995) 

have noted the linear nature of the damage, and Schmidt and others (1995) noted the 

apparent clustering of damage, with 54% of the damage occurring in the Los Gatos 7.5’ 

quadrangle. What is apparent in the density plot, and perhaps less apparent in figure 2, is 

the magnitude of the clustering relative to the linear concentration of damage. The 

cluster of data near Los Gatos is by far the dominant signal in the point pattern, with the 

density of points almost four times larger than the density along the linear concentration 

of damage. The cluster is at the southwest end of the Los Gatos embayment, a concavity 

in the range front near the town of Los Gatos (figure 2). In addition to lineaments, 

several mapped faults are also present within the region of the cluster. Is the cluster of 

damage due to local faulting, or another process, such as the shape of the embayment, 

which perhaps concentrated and focused seismic waves? The interpretation of the 

statistical results depends on the answer to such questions, and the questions are typically 

not easy to answer. If the clustering in the damage data (near Los Gatos) is due to a 

process independent of that which is potentially generating the lineaments, then the 

clustering will bias the resulting statistical analysis. 

In order to account for the possible bias of the cluster, it was modeled as a circular anomaly 

and the effect removed from the data. This was accomplished by examining the 

characteristic location and shape of the cluster observed in the density plot. The density plot 

shows the cluster of damage has a locus within the town of Los Gatos. Furthermore, the 

cluster is a roughly circular phenomenon that appears almost isolated from the rest of the 

damage. This can be seen in a cross-section of the density plot of figure 8 (top) that transects 

the cluster (figure 9). A gaussian curve fit to this cross-section models the cluster well to two 

 

1

 Density plots in one dimension are “smoothed” histograms, created by counting the number of data points 

within a moving window (rather than within a fixed bin as the histogram does), and applying a weighting 

function to the points so that points at the edge of the window contribute less to the total count within the 

window. In two dimensions the moving window is in the shape of a circle, typically tapering at the edges 

using a gaussian or quadratic kernel. See Silverman (1986) for a complete description of density plots. 

13
standard deviations. The model indicates that the cluster has a center 

14
100 350 600 850 1100 1350 1600 1850 2100

Distance across feature, meters

0

20

40

60

80

100

120

Figure 9. Circles are values from the density map taken along a cross-section. Black line is a gaussian curve (mean 1100, 

standard deviation 250) fit to the data. Note that the gaussian curve fits the data well to two standard deviations, indicating 

damage in this area forms a cluster, and little effect is seen beyond two standard deviations (grey vertical lines). 

Concentration of damage, number per 400 meter radius circle

within the town of Los Gatos2

 and a radius of about 500 m. There are 310 points within 

this area, and they seem to be attributable exclusively to the cluster, with little effect seen 

outside the model’s radius. 

The 310 points within a 500 m radius of the center of the modeled cluster were 

removed, generating a dataset referred to for the remainder of this paper as the 

declustered dataset. The density plot of the declustered dataset is shown in figure 8 

(bottom). The resulting dataset should be free of the effect of the dramatic clustering in 

the town of Los Gatos. However, it is possible that the clustering and the linear 

concentrations of damage are related to the same process as the lineaments, and that 

using the declustered dataset will reduce the sample size and introduce some bias. Since 

the nature of the cluster is not clear, both the original dataset and the declustered dataset 

will be examined in this report, and the results compared. 

RESULTS: Association of damage with lineaments 

As a first step, the point data is assumed to be homogenous over the lineament 

study area. To numerically approximate a random distribution of locations assuming a 

homogenous process (the probability is the same throughout the study area), the 

 

2

 coordinates are (590285, 4120396) UTM zone 10 NAD27 

15
lineament study area was rasterized at 5 m, and the distance of the center of each raster 

cell to the nearest lineament was recorded. This created a distribution against which to 

compare the observed distribution of damage. 

In general the shape of these histograms is asymmetric and skewed to the right. 

This is caused by two competing effects: the increase of area with distance away from an 

object (the area between 1 and 2 radii from a point, for example, is less than the area 

between 2 and 3 radii from the point), and the confines of the study area. Thus these 

histograms will increase until the restrictions imposed by the study boundaries limit the 

area available at large distances from the object. For a circular study area and a single 

point at its center, the histogram of the area at a given distance from the point will 

increase until the radius is reached. For an irregular study area with multiple objects, the 

amount of area available within the study area (typically) gradually tapers off with 

increasing distance from the objects. 

Figure 10 shows the histograms and density plots of the random and observed 

distributions. The median of the observed distribution, approximately 119 meters, is less 

than half the median of the random distribution, which is 240. Also note that the 

frequency of observed distances falls off much more rapidly towards the asymmetrical 

tail of the distribution than the random distribution. From this one can infer that the 

damage points are likely to be associated with the lineaments. In general, the damage 

points are simply closer to the lineaments. 

A more formal statistical test can be performed to demonstrate that the 

distributions are different. Figure 11 shows the cumulative distributions for both the 

random and the observed data sets. Again, assume that the points within the damage 

dataset are independent of each other; and that there is no tendency of clustering or 

avoidance among the points themselves. Any spatial association between the damage 

and lineaments is interpreted to be due to external influences, and in this case tectonic 

processes are the likely candidate. A two-sided Smirnov test, which is based on the 

difference between the two cumulative distributions, confirms the two distributions are 

different at greater than the 99% confidence level. From the data it can be seen that the 

observed damage points are more closely associated with the lineaments than the random 

dataset, thus the two tend to be spatially related. 

Now consider the inhomogeneous case, where damage is assumed to be 

homogeneous only within the area covered by streets. In the previous analysis it was 

assumed that the entire lineament study area was homogeneous, but this is not the case 

since the damage dataset only records damage in areas covered by streets. To take this 

into account the null hypothesis must be developed excluding the sub-areas that cannot 

contain damage to pavement and pipes breaks. 

Using the approximation of the lineament study area provided by the scanned 

USGS map discussed in the previous section, the same calculations (finding the distance 

to the nearest lineament for each cell) were performed. The result is the distribution of 

the null hypothesis of spatial randomness under the condition that only areas covered by 

streets can contain a damage point. Figure 12 compares the two null hypotheses of 

randomness, that considering the entire study area and that considering only the area 

covered by streets. This comparison shows the difference between the two null 

hypotheses of complete spatial randomness, the first which considers the entire study 

area, and the second which considers an area restricted to the streets that occur within the 

16
0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

observed distribution

0.000

0.001

0.002

0.003

0.004

Figure 10. Histogram of the random and observed distributions of the 

distance to the nearest lineament, in meters. Dashed lines show the 

median.

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

random distribution

0.000

0.001

0.002

0.003

0.004

17
0 500 1000 1500 2000

0.0 0.2 0.4 0.6 0.8 1.0

Figure 11. Cumulative distributions for the distance to the nearest lineament for a random set of points (solid, lower curve) and for the observed 

damage dataset (dashed, upper curve), for the lineament study area.

cumulative distance to nearest lineament

distance (meters)

study area. The median of the latter distribution is 249, and the histograms are seen to 

match closely. Figure 13 shows a quantile-quantile plot comparing the distributions. In 

this case the difference between the two distributions is not great. Apparently the streets 

offer a robust sampling of the lineament study area, and the change to the distribution of 

the null hypothesis is minimal. Therefore the comparison of the observed damage to the 

null hypothesis of complete spatial randomness performed in the previous section is 

unchanged. 

The previous analysis of comparing the observed damage to a null hypothesis of 

complete spatial randomness assumes that the cause of the damage is related to processes 

causing the lineaments, namely faulting along the range front of the Santa Cruz 

Mountains. If the large cluster of damage near Los Gatos, discussed in the previous 

section, is caused by a different process, say the focusing of seismic waves, then damage 

in this cluster could be adding bias to the statistical results. To address the question of 

whether or not the cluster is biasing, or even driving, the analysis, the analysis was 

repeated using the declustered damage dataset. 

A repeat of the analysis confirms the original results, although the observed 

distribution for the declustered dataset is not quite as sharply peaked, as shown in the 

histograms in figure 14 where it is compared with the distribution for the homogeneous 

lineament study area. The median has shifted from 119 meters to 154 meters. This 

indicates that the clustering does have an effect. The median of the declustered data set is 

18
0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

inhomogeneous distribution

0.000

0.001

0.002

0.003

0.004

Figure 12. Histogram plots of the homogeneous (entire study area) and

 inhomogeneous (area covered only by streets) distributions of the distance 

to the nearest lineament, in meters. Dashed lines are medians.

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

homogeneous distribution

0.000

0.001

0.002

0.003

0.004

19
0 500 1000 1500 2000

distance to nearest lineament, entire study area

0 500 1000 1500 2000

distance to nearest lineament, streets only

Figure 13. Quantile-quantile plot comparing the distribution of the homogeneous (entire study area) and inhomogeneous (streets only) distributions of the 

distance to the nearest lineament within the lineament study area. The distributions are very close to one another.

still closer to the lineaments than a randomly located set of points, but less dramatically 

so. The cumulative curves are shown in figure 15, and a two-sided Smirnov test again 

confirms that the distributions are significantly different from each other at the 99% 

confidence level. The damage points are located more closely to the lineaments than a 

randomly located set of points, even after the clustering effect has been mitigated and the 

inhomogeneity has been accounted for. 

Investigating the lineaments by treating them as zones 

Previous paragraphs have treated the lineaments as independent, individual lines. 

If instead one wishes to treat the lineaments as representing zones, then one must first 

convert the lineaments to zones, areas over which the process that generated the 

lineaments operates. Once zones are identified, points can be compared with them. 

Reasons for treating the lineaments or damage as zones might be that they are thought to 

be an expression of a stochastic process that operates over a zone, for which the existing 

lineaments or damage are one realization of that process. 

To convert the lineaments to zones, one must first decide how to define the zones. 

A simple way would be to specify some distance, d, from the lineaments, less than which 

is defined as within the lineament zone. One could then buffer the lineaments at the 

distance d to create the polygonal zones. The damage could be examined to see if a 

majority was located within the lineament zones. However, this is a binary version (only 

two options are considered: less than distance d, and greater than distance d) of the 

20
0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

observed distribution

0.000

0.001

0.002

0.003

0.004

Figure 14. Histogram plots of the random and observed distributions 

of the distance to the nearest lineament, in meters, after removal of the 

damage cluster. Dashed lines show the median.

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

random distribution

0.000

0.001

0.002

0.003

0.004

21
0 500 1000 1500 2000

0.0 0.2 0.4 0.6 0.8 1.0

Figure 15. Cumulative distributions for the distance to the nearest lineament for a random set of points (solid, lower curve) and for the observed 

damage dataset (dashed, upper curve), for the declustered lineament study area.

cumulative distance to nearest lineament

distance (meters)

original problem addressed in the previous section, where the entire range of distances of 

damage from the nearest lineament was considered. The binary case is a simplified 

version that considers only a single distance, d. Therefore if one defines the zones by 

means of a buffer, nothing is gained over the original analysis. 

An alternative to buffering the lineaments is to calculate the lineament density, as 

was done for the damage density in the previous section. In the case of linear features, 

the total length of features within a specified window is calculated across the map. The 

units are length per area. Again, the analyst must determine the size of the window. The 

density of lineaments in the lineament study area was determined using a 400 m window 

and a cut-off density of one standard deviation above the mean lineament density. This 

was chosen to satisfy the author’s judgment that the lineament zones should surround 

areas with abundant lineaments, yet not extend too far beyond the outer edge and into the 

adjacent empty space. Larger window sizes generate zones that extend beyond the 

concentration of lineaments into adjacent empty space, and smaller window sizes do not 

adequately combine dense areas into zones. Figure 16 shows the zones of lineaments 

defined as areas that are above the mean density of lineaments in the study area (top left), 

and as areas that are one standard deviation above the mean density of lineaments in the 

study area (top right). The first definition is compared with a buffer of 200 m from the 

lineaments (bottom left). This definition of a zone and the buffer are very similar, and 

are both judged to extend too far beyond the outer lineaments in the lineament groupings. 

The second definition of a zone, all areas one standard deviation above the mean 

lineament density, is deemed superior, and is shown with the lineaments (top right) and 

22

24
25
the damage (bottom right) overlain. While the zones are ultimately defined based on the 

judgment of the analyst, the procedure used to define the zones is quantitative and 

repeatable. This allows for a quantitative comparison between datasets, something that is 

not possible if the procedure used to define the zone boundaries were based solely on 

judgment (as in a visual assessment). 

The area where the density of lineaments is greater than one standard deviation 

above the mean within a 400 m moving window defines the zones of lineaments within 

the lineament study area. The area within the lineament zones makes up approximately 

31% of the total lineament study area. The zones can now be compared with the damage 

data. Since, as determined in the previous section, the streets seem to offer a robust 

sample of the lineament study area, the complication of using an inhomogeneous study 

area is not considered in the following analyses. 

One would expect that with 31% of the area covered, about 31% of the points 

should fall within the covered area. We can use the binomial distribution to test the 

hypothesis that the points are randomly distributed in space. Considering all damage that 

falls within the lineament study area, 652 out of 1046, or approximately 62%, of damage 

points fall within the lineament zones (figure 17). The probability that at least 652 out of 

1046 points fall within the damage zone is essentially zero (less than 4.5 x 10-14), thus we 

must reject the hypothesis that the damage is randomly distributed. The damage clearly 

shows a spatial association with at least some of the zones defined by the lineaments. 

The same analysis can be performed using the declustered dataset, to mitigate the 

influence of clustering processes on the analysis, as discussed previously. For the 

declustered lineament study area, the lineament zones make up 30% of this area. In the 

declustered dataset there are 353 out of 735 damage points, or approximately 48% of the 

points, that fall within the lineament zones. This leads to 222 expected points within the 

lineament zones under the assumption of complete spatial randomness (based on the 

binomial distribution). Again, the probability of 353 or more damage points falling 

within the damage zone is essentially zero (less than 6.4 x 10-14), and the damage clearly 

shows a spatial association with at least some of the zones defined by the lineaments. 

One can also compare the lineament zones with zones of damage generated from 

the declustered dataset, where the zones for the damage are defined using the same 

parameters: a 400 m search radius, and a cut-off of one standard-deviation above the 

mean. The zones are overlain in figure 18. The lineament zones make up approximately 

30% of the lineament study area, while approximately 53% of the area of the damage 

zones overlies the lineament zones. While a formal statistical analysis of the probability 

of the proportion of irregular shapes overlying one another within an irregular area is 

beyond the scope of this paper, a few qualitative observations can be made regarding the 

overlapping zones. The first is that 53% overlap seems rather significant, because under 

complete spatial randomness for a set of points one would expect about 30% overlap. 

The second observation is that the area of overlap shown in the inset seems to be the most 

significant overlap in the lineament study area. No clear linear pattern exists across the 

study area to suggest damage follows along a particular set of lineaments that could be 

inferred to be a fault. Rather, the zones overlap in patches, with perhaps the suggestion 

of a weak linear trend from southeast to northwest. 

Key decisions in the flow of the analysis in this section are based on the judgment 

of the analyst. These judgments include optimal window sizes and cut-off values used in 
generating the density plots, and the method used to decluster the data. In this sense the 

analyses are exploratory in nature. However, as emphasized, the quantitative procedures

defined for the analyses provide a quantitative foundation for repeating, and therefore 

comparing, analyses among different datasets. Furthermore, seeking a quantitative 

procedure by which to define fuzzy concepts such as “lineament zones” or “clusters of 

points” helps focus attention on the logic behind the scientific intuition that is often used 

to guide analysis. It is hoped that the procedures discussed have this effect. 

 

METHOD: comparing the damage data to complete spatial randomness, and looking 

for alignment of damage 

The previous section has suggested that the damage is preferentially located in 

zones similar to those defined by mapped areal photographic lineaments. The following 

sections are meant to present general methods of analyzing a point pattern, of which the 

damage data is one example. The question of randomness of the damage data will be 

explored without considering the previous results so that these new methods may be 

illustrated. 

The map pattern of the damage in the Santa Clara Valley from the Loma Prieta 

earthquake appears to be concentrated in a linear band outboard of, and parallel to, the 

range front of the Santa Cruz Mountains. There are also quite a number of damage points 

scattered throughout the damage study area. Is there a quantitative test that can be 

brought to bear that rules out the criticism that the damage points are randomly located? 

Can linear structure be quantitatively defined within the scattered damage points? 

To answer these questions a model of randomness (null hypothesis) must be 

constructed against which to compare the observed data. For a sample of points 

distributed over an area, what is meant by an alignment of points? One interpretation is 

that line segments defined by pairs of points tend to be aligned preferentially. Points 

clustering about an imaginary line would result in pairs of points aligned in a particular 

direction over a range of distances, whereas points scattered about an area would result in 

pairs of points with no particular alignment, given the constraints of the study area. 

Figure 19 shows two synthetic datasets defined over a 1 km by 1 km region. The first 

point set consists of ten points randomly located within the area, defined by combining 

ten random numbers, from 0 to 1, for the x-coordinates with ten random numbers, from 0 

to 1, for the y-coordinates. The second point set consists of ten aligned points, defined by 

combining ten random values, from 0 to 1, for the x-coordinates and assigning the ycoordinates according to the equation y = 0.6x+0.2 (an arbitrary line within the study 

area). Each pairing of points within the study area defines a vector, and the distribution 

of the directions defined by vectors, referred to from here on as direction vectors, can be 

used to examine structure within a point set; namely, whether or not the points are 

randomly distributed throughout the study area, and whether or not other structure can be 

detected. Direction vectors are simply the vectors defined by the pairs of points in a 

point set, and have length and direction defined as for any vector. Direction, in this case, 

refers to the compass direction in which the vectors point; the head and tail of the vectors 

are less significant for this analysis. In this paper the focus is on the orientation 

component of the direction vector and not the length. 

27
0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

distance (kilometers)

Figure 19. Location of 10 randomly placed points (left) and ten points placed randomly along a line (right) according to y=0.6x+0.2 (an arbitrary 

line). The direction vectors connecting the random points will not show a preferential direction; the direction vectors connecting the aligned points 

will.

random

distance (kilometers)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

distance (kilometers)

aligned

 5o 

tion. 

 of 

dy area were well within each others’ respective q05 and q95 (quantile) 

interva

s 

ween -90 and 90 (northerly) are considered, of which there 

are n(n

Figure 20 shows the distribution of the direction vectors for both the random point set 

and the aligned point set shown in figure 19. The distribution has been quantized into

bins and the bin mid-points connected by lines instead of using histogram bars. This 

emphasizes the shape of the distributions, and allows them to be added together easily. 

Note how the random point set results in direction vectors distributed roughly uniformly, 

whereas the aligned point set produces direction vectors that are aligned in one direc

Strictly speaking, the study area must be circular for the random distribution of the 

direction vectors to be perfectly uniform, but practically the deviation from a uniform 

distribution caused by a square study area is less than 5 percent. A Monte-Carlo test

100 simulations and 2000 points revealed that distributions of both the circular and

square stu

l. 

For a given point set with n points, and defining the direction vectors as any 

pairing of points other than a point with itself, there are n(n-1) total direction vectors. 

Half of these direction vectors are equivalent but of opposite sign e.g. direction vector

for points P1 and P2 are defined as P1 – P2 and P2 – P1. For some analyses it may be 

useful to consider the full set of direction vectors. Here north is defined as 0 degrees, and

only the direction vectors bet

-1)/2 in the point set. 

28
A set of randomly scattered points will, more or less, fill a study area. The shape 

of the study area, however, may limit the amount of scattering possible for a set of 

randomly located points. The direction vectors will therefore be biased by the shape of 

-80 -60 -40 -20 0 20 40 60 80

compass direction

0

10

20

30

40

50

60

70

80

90

aligned 

random

Figure 20. Frequency of direction vectors, in 5 degree intervals of compass direction, for the random point set and the aligned point set. The direction 

vectors of the random point set show a roughly uniform distribution, whereas the direction vectors of the aligned point set show a peaked distribution 

in the direction of the line along which the points fall. frequency

for 

use any irregularly 

shaped

oints 

peat the 

any times and average the result. This approach involves some 

program

n 

 

involves finding the probability density function for a fixed point paired 

the study area. A circular area containing randomly scattered points produces no 

preferential direction for pairs of points, on average, and the resulting distribution 

pairs of points is the uniform distribution. However, for an irregular area the random

distribution of direction vectors is not so straightforward. This is beca

 area may afford the opportunity for more points in one sector, biasing the 

resulting direction vector distribution. For irregularly shaped areas methods must be 

developed for defining the random distribution of direction vectors. 

One way to accomplish this is by Monte Carlo simulation: generate random p

within the study area, measure the distance and direction for all pairs of points, re

experiment m

ming and a fair bit of computer time, but has the advantage of producing an 

experimentally robust probability distribution where confidence intervals can be 

estimated. 

Another approach is to calculate the probability using a numerical approximatio

of the theoretical solution. This approach is advantageous because it is straightforward

and rapid once the method is understood. This approach is best explained in two parts.

The first part 

29
with each member of a randoml located point y set within a study area, and the second 

involves combining many of these into an overall probability density function for the 

entire area. 

Consider a fixed point, P1, within the study area (figure 21). Within the study 

boundary there is a fixed amount of territory that is between angle θ and angle θ + h

from P1. This territory consists of two wed ped areas opposite each other eithe

de of 

ge-sha r 

si . A point,

on

P1 Pi , added to either of these wed eg -shaped areas would result in a 

direction vector, defined by the point pair ( ) PP j , 1 , having an angle of between θ and 

θ + h, θ ( 1 , j) θ +<∠< hPP . If the second point, Pj , is added to the study area at random, 

it is as likely to land at one location within the study area as any other. Therefore the 

probability of a point landing in the area defined by the wedges is equal to the area of t

wedges divided by the total area under study. This is shown in figure 21 as 

he 

A

a1 , where 

is the combined area of both wedges and A is the total study area. Dividing the study 

1 a

area up into n n wedges for desired slices of h degrees each and recording the ratio of

A

aj

for each point pair ( ) Pj P1 , will result in a probability density functio for the f xed point, n i

30
P1. 

This probability density function can be numerically approximated by estimating 

the area of each wedge. This is accomplished by rasterizing (representing the study are

by a finite number of pixels) the study area and counting the pixels (which have a known

area) between 

a 

θ and θ + h degrees from P1, for all cells in the raster dataset. So, by 

rasterizing the study area, a probability density function for the angle from a fixed point, 

P1, can quickly be constructed by comparing P1 to every other cell in the raster datase

The second part of the explanation extends this idea to all points in the study a

Let each raster cell (pixel) within the study area represent a point, denoted i

t. 

rea. 

P or Pj . For 

each point, Pi , consider the direction vectors for all paired points ( ) ≠∋ PPPP jiji, , in the study ,

area. This yields the probability density function for the angle of a direction vecto

created 

r

by adding a new point,Pi . Summing across all points Pj within the study area 

yields the probability density function for the angle of direction vectors created by all 

pairs of a set of random points ( ) PP ij , within the study area. 

An approximation of the probability density function for the direction vec

pairs of random points within the irregular study area ca

tors of 

n be constructed by rasterizing 

31
the study area at the desired level of precision and calculating the directions for all pairs 

of points. Since each raster cell has the same area, the units cancel and one need only 

consider the number of points within a given direction to the total number of points 

within the study area. 

Now that a method exists to generate a probability density function of the 

direction vectors for a randomly located set of points within an arbitrarily-shaped area, 

the resulting theoretical distribution can be compared to an observed distribution. Since 

both distributions are circular and non-gaussian, a Chi-square test based on circular ranks 

test can be employed to see if the distributions differ significantly (Fisher, 1995). 

RESULTS: comparison to complete spatial randomness and alignment of 

damage

To test whether the damage points are non-randomly located, a distribution of the 

direction vectors for randomly scattered points was compared to the empirical 

distribution of direction vectors for the study area. This test is comparing the damage 

data with complete spatial randomness, so the entire dataset, rather than the declustered 

dataset, should be used. The reason for this is that the relation of damage to one or more 

processes would legitimately indicate a departure from random behavior; additional 

processes controlling damage locations does not affect the outcome of the test. The 

distribution of direction vectors for a random set of points was constructed using the 

numerical approximation method discussed in the previous section. A histogram of the 

direction vectors, with a density line superimposed, is shown in figure 22 (bottom). Note 

how the shape of the study area causes a preferential alignment in the northwest 

direction. 

The distribution of direction vectors for the observed damage locations are shown 

in figure 22 (top) as a histogram and density plot. There are over one million direction 

vectors for the sample of 1427 damage locations. Note that the two histograms differ 

somewhat in shape, but more importantly the mode in the random distribution is 

displaced from the mode in the observed distribution. 

32
0.015

0.000

0.005

0.010

-90 -60 -30 0 30 60 90

-90 -60 -30 0 30 60 90

compass direction (degrees from north)

0.000

0.005

0.010

0.015

Figure 22. Combined histogram and density plots (black lines) for the observed distribution of direction 

vectors (top) and the theoretical distribution of direction vectors (bottom). Note that the observed 

distribution is multimodal, whereas the theoretical distribution is (largely) unimodal. The y-axis is base

on the density plots, which normalize the area under the density curve to one.

d 

33
Figures 23 and 24 compare the distributions to each other, using a quantilequantile plot and cumulative plot, respectively. As the quantile-quantile plot (figure 23) 

shows, the two distributions differ noticeably, with the observed distribution having 

much fewer direction vectors in the range of -90o to about -50o than the random 

distribution. This difference also stands out on the cumulative distribution plot (figure 

24), which is the distribution that is used for the Chi-square test based on the circular 

ranks test to compare two distributions. Judging visually, there is a clear distinction 

between the two distributions. 

-90 -60 -30 0 30 60 90

empirical distribution of direction vectors, degrees

-90 -60 -30 0 30 60 90

theoretical distribution of direction vectors, degrees

Figure 23. Quantile-quantile plot comparing the observed and theoretical random 

distributions formed by the direction vectors. 

The non-parametric Chi-square test based on circular ranks was performed to 

quantify the difference between the two distributions (Fisher, 1995). The test showed a 

highly significant difference, beyond the 99% confidence level. However, such tests can 

be sensitive to very large sample sizes. For example, Rock (1988) points out that the 

significance levels for large (>100) samples are approximated for the Smirnov test, the 

non-parametric test used to distinguish two different distributions in the previous section, 

so supporting evidence for the degree of difference between the distributions would be 

helpful. 

A Monte Carlo simulation that placed points within the study area at random 

locations was performed to test the veracity of the numerical approximation method and 

to confirm the results indicated by the Chi-square circular ranks test. 1427 points, 

equivalent to the sample size of the damage observations, were distributed at random 

locations throughout the study area, and the distribution of the direction vectors 

calculated. 100 simulations were performed, and the average, 0.05, and 0.95 quantiles 

recorded at five degree intervals. Figure 24 displays the results of the Monte Carlo 

34
-90 -60 -30 0 30 60 90

trend of direction vectors, degrees

0.0 0.2 0.4 0.6 0.8 1.0

Cumulative distribution

theoretical 

mean monte carlo

0.95 quantile

0.05 quantile

observed

Figure 24. Cumulative distributions of the direction vectors for the theoretical random, mean monte carlo random (including the 0.05 and 0.95 quantile 

confidence intervals), and the observed damage data set. The mean monte carlo and theoretical random distributions are almost indistinguishable in this plot.

35
051

trend of direction vectors, degrees

0

0.70 0.71 0.72 0.7

Cumulati

3 0.74 0.75 0.76

ve distribution

theoretical 

mean monte carlo

0.95 quantile

0.05 quantile

Figure 25 (close-up). Close-up of figure 25, showing that the difference between the theoretical distribution calculated numerically is close to the mean 

monte-carlo simulation, and within the 0.05 and 0.95 confidence envelopes.

simulation, including the 0.05 and 0.95 quantiles that serve as confidence envelopes. The 

distribution of the direction vectors for the average Monte Carlo simulation and the 

numerically calculated distribution are indistinguishable in the figure, and upon close 

inspection show only very minor deviations from each other (figure 25 shows a close-up 

of the graph, to better distinguish the numerically-calculated theoretical distribution of 

complete spatial randomness and the mean Monte Carlo distribution of complete spatial 

randomness). This indicates that the numerical solution is a viable alternative to Monte 

Carlo simulation for this type of analysis. The 0.05 and 0.95 quantile confidence 

envelopes are also quite close to the average distribution, indicating that there is little 

variation within the random distribution of direction vectors, at least with the sample size 

given. The observed distribution is clearly outside of the 0.05 and 0.95 confidence 

envelopes, supporting the results of the Chi-square circular ranks test. The damage data 

is clearly not randomly distributed. 

Three potential problems with the investigation must be mentioned. The first is 

preferential sampling due to the limitation of damage being confined to public areas, 

primarily streets. Since streets are commonly arranged in grids, there is the danger of 

sampling finely parallel to the streets and coarsely at an angle to them, introducing 

anisotropy into the sampling process. However, a street map of the study region 

revealed no such systematic pattern to the street network, and the previous investigation 

showed minimal bias in the lineament study area. The study area is large enough to 

encompass several communities, each with separate street patterns that do not coincide in 

direction with each other. A related problem is that sampling along streets can introduce 

a linear pattern in the data for small local areas. This is noticeable within the large cluster

describe in the previous section. The third potential problem is the possible association 

of damage with surface geology. While there is undoubtedly a relationship between the

36
two, the surficial geology is also closely related to the tectonics of the area, with alluvial 

fan units forming along the mountain front and giving way to broad mudflats towards the 

shore of the bay. This creates a trend in the spatial arrangement of the Quaternary 

geologic units. The damage points do not appear to be preferentially associated with any 

particular geologic unit. Indeed, they seem to cut across the grain of the Quaternary 

geology. Since there is no obvious relationship between the two independent of the local 

tectonic influence, and the two likely co-vary because of this, it is assumed that the 

Quaternary geology has an insignificant effect on the location of the damage. 

The results indicate that the observed distribution of damage in the Santa Clara 

Valley differs from that of randomly distributed locations. Furthermore, the observed 

distribution shows distinct peak at the mode, 40o northwest (figure 22). This peak is 

narrower than the broad mode of the random distribution. This suggests preferential 

alignment of points at 40o northwest. 

The numerical method used to test the null hypothesis of a random distribution of 

direction vector orientations is a viable alternative to Monte Carlo simulation, and may be 

preferred because it is easy to calculate. 

DISCUSSION: Development of a process model and a method of spatial filtering 

Visually the pattern of damage in Santa Clara Valley seems to be preferentially 

aligned in a northwest direction along a single, major line near the range front, with 

perhaps some smaller, secondary alignments further from the range front, and some 

scattered points throughout the study area (figure 2). The density plot (figure 8, top) also 

shows that a significant cluster appears in the dataset near the town of Los Gatos and the 

distribution of direction vector orientations for the damage locations also shows a 

prominent spike at approximately 40o northwest. It appears at least three patterns are 

combined within the damage data. The damage forming the cluster pattern was separated 

previously. Is there a quantitative model that can be applied to the dataset to separate the 

declustered damage into two subsets, one representing a linear pattern and the other a 

random pattern? 

 Visual inspection of the declustered damage, and the distribution of the resulting 

direction vector orientations, suggest a simple end-member model for damage, namely 

that damage is either aligned in a northwest direction, or scattered randomly throughout 

the study area. This combination of patterns would result in the direction vectors being 

defined by a mixture of pairs of random points, pairs of aligned points, and pairs of one 

random and one aligned point. 

To investigate this behavior, consider the synthetic dataset of random and aligned 

points from figure 19, combined into one plot in figure 26. 

37
aligned

random

0.8

1.0

0.6

distance (kilometers)

0.0

0.4

0.2

0.0 0.2 0.4 0.6 0.8 1.0

distan

 placed poi

ce (kilometers)

Figure 26. Location of 10 randomly nts and ten points placed randomly along a line according to 

y=0.6x+0.2 (an arbitrary line). The direction vectors connecting the random points will not show a preferential 

direction; the direction vectors connecting the points along the line will.

The direction vectors for this point se will re total of 190 dire n vectors 

(20*19/2), 45 of which are generated by pairings of aligned points (10*9/2), 45 of which 

are generated by pairings of random points, and 100 of which are generated by pairing a 

random point with an aligned point. Figure 27 (top left) is equivalent to figure 20, and 

t sult in a ctio

38
Frequency of direct

 and random 

ion vectors for aligned 

cases

-80 -60 -40 -20

compass

0 20 40 60 80

 direction

05

101520253035404550

-80 -60 -40 -20 0 20 40 

05

101520253035404550

Frequency of direc n vectors for a

 in idually

tio 

div

compass direction

-80 -6 -4 -20 0 60

comp s direction

0 0

as

20 40 80

05

101520253035404550

Frequency of direction 

 individually a

vectors for all cases,

nd combined

alignedrandomaligned & randomall direction vectors

Figure 27. The direction vectors for a set of comand random points are shown here broken down components: aligned-aligned and random-randovectors (upper left), then adding the aligned-ranvectors (upper right), then all c ponents with tof direction vectors (bott m left Note how the dire on vectors pattern.

om 

o ). 

cti show a random 
figure 27 (top right) adds the direction vectors for the random-aligned pairings. The 

direction vectors in figure 27 (top right) add together to create the distribution of 

direction vectors for the whole point set, shown in Figure 27 (bottom left). Note that the 

distribution of direction vectors for the random-aligned pairings is roughly uniform. One 

way to think of this result is that a single point, when added to a random point set, will 

produce direction vectors that appear random. The direction vectors for the randomaligned pairings is the combined direction vectors for each aligned point with the random 

point set. 

One can consider the direction vectors associated with the pairing of the aligned 

points as a signal, and the rest of the direction vectors as noise. This can be defined by a 

noise factor, k, where is the number of aligned points, and therefore 

1)

is the total number of points in the point set. In general, increasing n will cause 

percentage of signal in the direction vector distribution to decrease towards 

2)

na

a = nkn 

2

1

k . 

so as the size of the point set increases, the proportion of direction vectors 

considered noise approaches 2

1 1

k − (see Appendix I for proof). If the signal-to-noise 

ratio (in this case the ratio of the direction vectors considered signal to the total number 

of direction vectors) can be estimated from the direction vector distribution, can be 

estimated by the relationship defined above: 

na

 , where ( )

( ) 1

1

−

− =

aa

aa

knkn

nn R

R

k 1 = 

and by 1), == k

n

na

R

n

1 3) 

where R is the proportion of direction vectors considered signal to the total number of direction 

vectors, is the number of aligned points, n is the total number of points in the set, and k is a 

constant. 

For large values of n the equation simplifies further, 

since 

na

( )

( )

( )

, 1

1

1

2

n

nn

knkn

nn R aa aa −

aa

≈ −

− =

then for large values of n
4)

( )

( ) 1

1

1 2 −=

−

≈ nn aa

nn

n

n

R

n 

aa

For example, consider the direction vector distribution for the point set above, as 

if the structure of the point set were unknown. The distribution suggests an alignment of 

points at north 55o east, and randomly distributed direction vectors otherwise. The mean 

frequency of the data, excluding the peak direction of 55o east, is 4.11 direction vectors 

per 5o. Assuming this applies to all directions, there are 148.11, or 148, direction vectors 

that are in the noise category, and therefore 42 direction vectors in the signal category. 

The ratio of direction vectors considered signal to the total number of direction vectors is 

therefore 42/190. By equation 3, the number of aligned points in the point set is 

42/190/20 =9.4, or 9 points. Considering the direction vector distribution for each 

individual point, the points with their direction vector mode at 55o east are logical 

candidates to be the aligned points. In this synthetic example there are 10 points with 

equal frequency of direction vectors for 55o east, with no clear way of distinguishing 

them. The model considers only the average random response, and has not accounted for 

the individual character of this dataset. In this synthetic example the number of aligned 

points is under-predicted because the model assumes that there is a random component to 

all directions. In this small sample, no random point fell in the aligned direction, and 

thus the random signal was absent in the aligned direction. Since the calculated number 

of aligned points is approximate, bringing additional information to bear on borderline 

cases would be the next step in any investigation. 

Clusters of points can also produce a peak in the direction vector distribution. 

Figure 28 shows the distribution resulting from taking the 10 random points in a 1 km by 

1 km area described above, and adding one point to the point data set at ten times the 

range (10 km away). In this modified dataset the random points behave like a cluster, 

and the added point is an outlier. As can be seen from the figure, the direction vector 

causes a peak in the distribution of direction vectors. Two clusters sufficiently distant 

from one another essentially form a line between each other, and therefore cause a peak 

similar to collinear points. As with the lineaments, clusters of points can the bias the 

results of the analytical method discussed in the previous paragraph. 

The previous example has shown that the direction vector distribution for a point 

set can

rate the components of the direction vector 

as been quantized i 5o bins 

 be thought of as different components adding together to create the total 

distribution. Given a direction vector distribution, one can construct simple models of 

linear structure and attempt to sepa

distribution. For a simple model of aligned points and random points, one can estimate 

the number of aligned points, and, using the direction vector distribution for each 

individual point, develop a set of candidate aligned points. In this case the direction 

vector distribution can be used as a filtering process. 

For the observed damage dataset, the declustered data is used to avoid the 

problem the large cluster in the town of Los Gatos will cause. Small clusters in the data 

will still have an effect, but the largest effect will be minimized. The histogram of the 

distribution of direction vectors for the declustered dataset is shown in figure 29 (smalldashed line; in this figure, as in figure 20, the distribution h nto

41
-80 -60 -40 -20 0 20 40 60 80

compass direction

0

2

4

6

8

10

frequency

-80 -60 -40 -20 0 20 40 60 80

compass direction

0

2

4

6

8

10

Figure 28. Distribution of direction vectors for the set of ten randomly located points and an

additional outlier point located at ten times the range of the random points. The distribution 

of direction vectors for the random set (thin solid line) is compared with that of the outlierrandom direction vectors (dashed line) in the top figure, and both are compared with the

direction vector distribution for the entire dataset (thick solid line) in the bottom figure.

frequency

and the bin mid-points connected by lines instead of using histogram bars, in order to 

emphasize the shape of the distribution and to be able to subtract one distribution from 

the other). It has a similar shape to the complete observed damage dataset. The largest 

number of random direction vectors possible, without exceeding the observed number of 

42
direction vectors at any given point, is fit to the distribution of direction vectors for the 

declustered dataset (figure 29, large-dashed line), and subtracted from it, to obtain the 

residual distribution (figure 29, solid line). The residuals have a clear peak from 55o to 

40o northwest. 

-90 -60 -30 0 30 60 90

compass direction, degrees from north (0)

0

20,000

00

43

40,0

60,000

80,000

100,000

Frequency of direction vectors 

declustered data

residuals

random data

Figure 29. Distribution of direction vectors for the declustered damage dataset, a random dataset, and the residuals of the distribution of the 

declustered data minus the distribution from the random data. 

was fit to the declustered distribution.

The largest subset possible of direction vectors forming the random distribution 

rection 

Figure 30 shows the results of choosing the damage points with their individual 

direction vector mode between 55o to 40o northwest. That is, for any point Pi , consider 

all direction vectors for that point: ,(),...,,(),...,,(),,(),,{( )} i 1 i 2 i 3 ji PPPPPPPPPP ni , 

excluding ) ,( PP ii , and select the mode of this distribution as the mode of the direction 

vectors for the point Pi . Figure 30 considers points whose mode is between 55o to 40o 

northwest. When separated into groups according to their mode, the alignment of points 

is moderately linear, strongly linear, weakly linear and somewhat clustered. The 157 

points having a direction vector distribution mode of 50o northwest, shown in the upper 

right of figure 30, show the strongest linear pattern. There are 51,751 residual d

vectors 

i

for this direction. Estimating the expected number of points from the direction 

vectors using a model of linear and random points leads to 51751 ≈≈ 227 a n , 

overpredicting the observed 157 points having a direction vector distribution mode of

by about 30%. If one considers the entire peak from 55o to 40o northwest, the direction

vectors must be added together in equation 4. The estimated number of direction vecto

 50o

rs 

for a peak signal from 55o to 40o northwest is 206377, out of a total of 1246572 direction 

vectors in the entire distribution. This results in an estimate of ≈ 206377 ≈ 454 a n . 

The total number of damage points with a mode from 55o to 40o northwest is 741. The 

discrepancy between observed and predicted aligned damage is probably because the 

damage is not perfectly aligned, as the model demands, but are more dispersed, which 

causes fewer direction vectors to be aligned in one direction. It therefore takes more 

points to generate the same number of direction vectors in the aligned direction. It also 

indicates that the model of a linear alignment of points plus a random set of points is not

completely applicable to the declustered damage dataset. This is most apparent for the 

oints with a direction vector distribution mode of 40o northwest, where outlier points in 

dy area can be seen responding to clusters of points in the 

souther is effect. 

st apparent within the damage data can be 

neaments. The lineaments are aligned in a 

northwest-trending direction (figure 3). If each straight segment of the mapped 

lineaments is considered as a vector (so that a single mapped lineament that consisted of 

many segments would be considered many vectors, connected head-to-tail), the mean 

direction of the lineaments can be found by calculating the mean direction of all of the 

vectors. The mean direction of the vectors can be found by adding the vectors and 

calculating the direction of the resulting sum. The component vector information is 

easily found by extracting the coordinate information from the GIS (appendix 2). The 

mean direction for the mapped lineaments is 49o northwest. Thus the prominent linear 

pattern within the damage data matches quite closely with the mean direction of the 

mapped lineaments, again suggesting a common factor, faulting along the range front of 

the Santa Cruz Mountains, links them. 

 

Problems with the model 

The method of fitting a model of aligned points plus random points to a scattered 

point dataset considers points aligning in a single direction. For practical application to 

faulting in geology, this may be somewhat limited, because faults may bend, or show 

more than one preferential direction. For investigation of points following a curving line, 

or for conjugate sets, the model for this simple analysis breaks down. In this particular 

area it is fortunate that nature seems to have fairly regular behavior. 

The method is also affected by the width of fault zones. A fault zone adds shortdistance noise to the distribution of direction vectors, and the resulting direction vector 

peak in the direction vector distribution is not as pronounced. This leads to an 

underprediction in the number of points producing the “signal”. 

Lastly, other structures in the scattered point dataset, such as clusters and outliers, 

or multiple clusters, can interfere with the signal of aligned points. These structures are 

neither linear nor necessarily random, and must be investigated separately. 

p

the northern portion of the stu

n portion. The declustering algorithm reduces, but does not eliminate, th

From the declustered data a clear linear trend emerges for 50o northwest, and a 

reasonably linear trend for 55o northwest. For the directions 45o northwest to 40o 

northwest the trend is less linear and includes some signs of clustering patterns. 

The linear trend of 50o northwe

compared with the mean trend of the mapped li
SUMMARY AND CONCLUSIONS 

discuss me object than would be the 

subset of points that are aligned. The techniques were demonstrated using a dataset of 

r 

objects (point, line, or polygon features), applying techniques previously developed in the 

the theo

for an a hypotheses of 

lineame

taset, such 

phenom

gaussia

bias present in the damage-lineament analysis. The results demonstrate that the cluster is 

cluster 

each an

method

compared with that developed using Monte Carlo methods, and the two were found to 

damage

compar

indicate

distribu modal, whereas the observed distribution of 

that a th union of a set of points aligned along one 

del for the patterns 

process dominates the signal of the entire dataset and that a linear signal can be seen 

when analyzing the declustered dataset, though some of the signal from clustering in the 

data is still present. The most significant linear trend is 50o northwest, where the model 

Three analytical techniques relevant to the analysis of point patterns were 

ed: (1) testing whether or not points are closer to so

case with complete spatial randomness, (2) testing whether or not a set of points 

exhibited complete spatial randomness, and (3) a method of filtering a point set for a 

damage caused by the Loma Prieta earthquake and a dataset of mapped lineaments within

the damaged area. 

The first model of random behavior defined the relationship of points to othe

field of urban planning (Okabe and Fujii, 1984). By applying basic probability theory, 

retical distribution of the distance a set of randomly located points to the objects, 

rbitrarily shaped study area, can be constructed and used as a null 

randomness for comparison with observed data. The first model of random behavior 

defined the relationship of random points to a set of linear features, in this case 

nts. 

Since the above analysis can be sensitive to inhomogeneities in the da

as clustering, a declustering process was proposed that models a cluster as a circular 

enon whose point density decays from the center of the circle according to a 

n function. When applied to the damage data, the declustering diminished the 

the most significant pattern in the data in terms of damage concentration and that the 

significantly influenced the results of the initial statistical analysis. 

The second model of random behavior defined the relationship of points to each 

other. The theoretical random distribution of direction vector orientations, defined by 

d every pairing of points in a set of points, was constructed using numerical 

s. This method of developing a random distribution of direction vectors was 

agree quite closely. 

The method outlined above was used to demonstrate that the locations of the 

 data are not randomly located throughout the study area. The distributions were 

ed using a Chi-square test based on circular ranks, and the null hypothesis of

randomness was rejected. The preferential alignment of the random process was found to

be about 60o northwest, whereas that of the observed process was 40o northwest. This 

s that the observed process is biased to the north. Furthermore, the random 

tion of direction vectors is uni

direction vectors is multimodal. 

The distribution of direction vector orientations for the damage data suggested 

eoretical model, composed of the 

direction and a set of randomly located points, was a reasonable mo

observed in damage data. Using this as a base model, it was discovered that the cluster 

46
predicted 227 points would be aligned in that direction, and in the observed damage 

 there were 157. The linear trend in the damage data of 50o northwest matches 

osely the mean trend of the mapped lineaments, which average 49o north

Numerical methods can be used to develop null models of complete spatial 

ness over (planar) irregularly shaped stud

dataset

quite cl west. 

random y areas. These models can be generated 

a 

set of p ction vectors resulting 

ligned. 

This fil dataset into component patterns of 

ACKN

The following people generously provided data and time for thoughtful 

discussions: Carl Wentworth, Kevin Schmidt, and Robert Jachens. Reviews by Donald 

Singer and Robert Simpson were very helpful. Special thanks to Michael Barall, who 

greatly improved the mathematical underpinnings of the methods. This work would not 

have been possible without their help. 

quickly using modern GIS software. They are applicable to determining if a set of points

is associated with some other set of objects (points, lines, or polygons) or determining if 

oints itself exhibits complete spatial randomness. The dire

from a point dataset can be used to filter a point dataset to find points that are a

tering process can be used to separate the point 

aligned and non-aligned points, provided a single linear pattern exists and clustering of 

points is not excessive. 

OWLEDGMENTS 

47
REFER

Brabb,

Diggle,

s, 

Cambridge. 

, Hecker, S., and Prentice, C.S., 2006, 

of Qua Bay region: U.S. Geological 

Hitchcock, C.S., Kelson, K.I., and Thompson, S.C., 1994, Geomorphic investigations 

f deformation along the northeastern m

Geological Survey Op

Hitchcock, C.S., and Kelson, 

Santa Clara Valley, San Francisco B

ic hazard a ecurrence: Geology, v. 27, no. 5, p.391–

., and Shelly, A., 1996, S+SpatialStats: User's 

anual for Windows and UNIX: Springer-Verlag, New York, 327 pp. 

Langenheim, V.E., Schmidt, K.M., and Jachens, R.C., 1997, Coseismic deformation 

during the 1989 Loma Prieta earthquake and range-front thrusting along the 

southwestern margin of the Santa Clara Valley, California: Geology, v. 27, p. 

387-390. 

McLaughlin, R.J., Clark, J.C., Brabb, E.E., Helley, E.J., and Colón, C.J., 2001, 

Geologic Maps and structure sections of the southwestern Santa Clara Valley and 

southern Santa Cruz Mountains, Santa Clara and Santa Cruz Counties, California: 

U.S. Geological Survey Miscellaneous Field Studies Map MF–2373, 13pp. 

Okabe, A., and Fujii, A., 1984, The statistical analysis through a computational method 

 in rela n to its surrounding network: Environment and 

14. 

Okabe, A., Fujii, A., Oikawa, K., and Yoshikawa, T., 1988, The statistical analysis of 

a distribution of activity points in relation to surface-like elements: Environment 

and Planning A, v.20, p.609-620. 

Rock, N.M.S., 1988, Numerical geology: Springer-Verlag, Berlin. 

ENCES 

 E.E., Graymer, R.W., and Jones, D.L., 2000, Geologic map and map database

of the Palo Alto 30’ x 60’ quadrangle, California: U.S. Geological Survey 

Miscellaneous Field Studies Map MF–2332, 32 p. 

Connover, W.J., 2003, Practical non parametric statistics: John Wiley & Sons Inc., 

584 p. 

 P.J., 2003, Statistical analysis of spatial point patterns: Arnold, London, 159 pp.

Fisher, N.I., 1995, Statistical analysis of circular data: Cambrdge University Pres

Graymer, R.W., Bryant, W., McCabe, C.A.

Map ternary-active faults in the San Francisco 

Survey Scientific Investigations Map 2919, map scale 1:275,000. 

o argin of the Santa Cruz Mountains: U.S. 

en File Report 94-187, map scale 1:24,000. 

K.I., 1999, Growth of late Quaternary folds in southwest 

ay area, California: Implications of triggered 

slip for seism nd earthquake r

394. 

Kaluzny, S.P., Vega, S.C., Cardoso, T.P

m

of a distribution of points tio

Planning A, v.16, p.107-1

48
Schmidt, Kevin M., Ellen, Stephen D., Haugerud, Ralph A., Peterson, David M., and 

f rangeuthwestern margin of the Santa Clara Valley, California: U.S. Geological 

Report 95-820, 33pp. 

Phelps, Geoffrey A., 1995, Breaks in pavement and pipes as indicators o

front faulting resulting from the 1989 Loma Prieta earthquake near the 

so

Survey Open-File 

Upton, G., and Fingleton, B., 1985, Spatial data analysis by example, Volume 1:Point 

pattern and quantitative data: Wiley, Chichester, England.

U.S. Geological Survey, 2007, October 17, 1989 Loma Prieta earthquake 

http://earthquake.usgs.gov/regional/nca/1989/

Wentworth, Carl M., Blake, M. Clark, Jr., McLaughlin, Robert J., and Graymer, 

Russell W., 1998, Preliminary geologic map of the San Jose 30 x 60 minute 

quadrangle, California: A digital database: U.S. Geological Survey Open-File

Report 98-795, 52p. 

49
50

APPENDIX I 

Given: the union of two independent sets of points defined on a plane, one a set 

of randomly located points, the other a set of points that fall along an arbitrary line, such 

that the total number of points in the union of the two point sets is n . Direction vectors 

are generated by the pairing of points, with one direction vector for each point-pair. The 

complete set of direction vectors is generated by the unique pairing of all points. 

Prove: the ratio of the number of direction vectors from the aligned point set to 

the number of direction vectors from the union of the two point sets approaches 2

1

k

 as n

approaches infinity.

Let na = the number of points in the aligned set 

a nkn == the number of total points in the union of the two sets 

( ) nn aa 2/1 =− the number of direction vectors in the aligned point set, 

here considered signal

( ) knkn aa 2/1 =− the total number of direction vectors in the union of the 

two point sets 

Proof: the ratio of direction vectors considered signal to the total number of 

direction vectors for a given point set is 

( )

( )

( )

( )

( )

( ) 1

1

1

1

2

1

2

1

−

− =

−

− =

⎟

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎜

⎝

⎛

−

−

a

a

aa

aa

aa

aa

knk

n

knkn

nn

knkn

nn

The limit of this ratio as n → ∞ is 

( )

( )

( )

( )

( )

2

2

1

1

1 1

1

1 1

1

1 lim lim lim

k

k

n

n

kkn

n

kknk

n

a

a

a n

a

a n

a

n

=

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

⎟

⎠

⎞ ⎜

⎝

⎛ −

− ⎥ =

⎦

⎤ ⎢

⎣

⎡

−

− ⎥ =

⎦

⎤ ⎢

⎣

⎡

−

−

∞→ ∞→ ∞→
51

APPENDIX II 

Technical specifications for various spatial analysis tasks 

 

Declustering point data 

1. Convert the (vector) point pattern to a continuous surface by generating a density 

map. 

2. Digitize a cross-section line across the anomaly (visible on the density map); add 

vertices to the line at regular sampling intervals. 

3. Convert the cross-section-line vertices to points; extract the value of the density 

map at the location of the points. 

4. Output these sampled values (Y values) and their respective distance along the 

cross-section-line (X values) to a statistical package and model the anomaly. 

Approximating the probability distribution of the distance to the nearest line for 

complete spatial randomness 

1. If the study area is in vector format, convert the study area to raster in order to 

generate an approximation of a space-filling set of points. 

2. Convert the raster (cells) to (vector) points. 

3. Find the distance to the nearest line for each point, which as a set represent the 

entire study area (e.g. the NEAR command in Arc/Info). Ensure that each record 

includes the line id number and distance. This will allow for further analysis using 

subsets of data based on the individual lines. 

4. Output the distance data to a statistical package. This distribution of distance is 

an approximation of the probability distribution of the distance to the nearest line for 

complete spatial randomness within the study area. 

Finding the observed probability distribution of the distance to the nearest line for a 

set of points

1. For the observed data, find the distance of each observation to the nearest line 

(e.g. the NEAR command in Arc/Info). Ensure that each record includes the line id 

number, point id number, and distance. This will allow for further analysis using 

subsets of data based on the individual lines. 

2. Output the distance data to a statistical package. This is the observed distribution 

of the distance of points to lines, and can now be compared with the approximation of 

the theoretical distribution. 

Finding the average orientation of lines based on individual line segments 

1. Consider each pair of vertices along the length of a line as a vector. 

2. Extract the coordinates of each vertex, in order, along the line, and calculate 

)( 12 −=Δ xxx and )( 12 =Δ − yyy for each pair. 

3. To add the vectors and find the direction of the average vector, calculate 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

Δ∑

− Δ∑

y

1 x

tan and convert the result to the compass direction.