Kfd Implementation

Environ Geol (2005) 47: 1083–1098 DOI 10.1007/s00254-005-1241-2

Y. Gao E. C. Alexander Jr R. J. Barnes

Received: 28 October 2003 Accepted: 12 January 2005 Published online: 19 April 2005
Springer-Verlag 2005

Y. Gao (&) Department of Physics, Astronomy, and Geology, East Tennessee State University, Johnson City, TN 37604, USA E-mail: [email protected] Tel.: +1-423-4394183 E. Alexander Jr Department of Geology & Geophysics, University of Minnesota, Minneapolis, MN 55455, USA E-mail: [email protected] R. Barnes Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55455, USA E-mail: [email protected]

ORIGINAL ARTICLE

Karst database implementation in Minnesota: analysis of sinkhole distribution

Abstract This paper presents the overall sinkhole distributions and conducts hypothesis tests of sinkhole distributions and sinkhole formation using data stored in the Karst Feature Database (KFD) of Minnesota. Nearest neighbor analysis (NNA) was extended to include different orders of NNA, different scales of concentrated zones of sinkholes, and directions to the nearest sinkholes. The statistical results, along with the sinkhole density distribution, indicate that sinkholes tend to form in highly concentrated zones instead of scattered individuals. The pattern changes from clustered to random to regular as the scale of the analysis decreases from 10–100 km2 to 5–30 km2 to 2–10 km2. Hypotheses that may explain this phenomenon

are: (1) areas in the highly concentrated zones of sinkholes have similar geologic and topographical settings that favor sinkhole formation; (2) existing sinkholes change the hydraulic gradient in the surrounding area and increase the solution and erosional processes that eventually form more new sinkholes. Keywords Karst Feature database (KFD) Æ Nearest neighbor analysis (NNA) Æ Nearest neighbor index (NNI) Æ Complete spatial randomness (csr) Æ Distance to nearest neighbor (DNN) Æ Minnesota

Introduction

Overview of spatial analysis on cave and karst studies

This paper is the second of a series of two papers describing the Karst Feature Database (KFD) of Minnesota (Gao et al. 2005). The development, implementation, and data analyses of the Minnesota KFD are described in detail in the first author’s Ph.D. dissertation (Gao 2002). This paper discusses the analysis of sinkhole distribution based on the sinkhole data stored in the Minnesota KFD. The fundamental scientific question of this study is what controls sinkhole distribution in Minnesota. A complete statewide KFD provides data and tools to test hypotheses about sinkhole distribution in Minnesota.

Karst scientists and researchers have developed and used several analytic methods in cave and karst studies. Lineament analysis and nearest-neighbor analysis (NNA) are two of the most commonly used approaches to study karst feature distributions. Lineament analysis Geological structures in karst terrain are often observable as traces or lines, commonly referred to as ‘‘lineaments.’’ Mapping and interpretation of lineaments

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in relation to sinkhole distribution have been used in many karst areas. Matschinski’s (1968) procedure was the first to determine the local alignment common to every three points (e.g. sinkholes) and then to plot the distribution of these local alignment directions on a histogram. He concluded from the interpretation of the histogram that local alignments of sinkholes are mainly controlled by tectonics of the adjacent regions in Lake Constance, Germany. Barlow and Ogden (1982) used a modified version of the Kolmogorov–Smirnov (K–S) test to compare joints, straight-cave segments, and photo-lineament orientations. They concluded that for low altitude photo-lineaments and straight cave-passage segments, the orientations of joints, straight-cave segments, and photolineaments are similar in Benton County, Arkansas. Using evidence from field work, interpretation of aerial photography, and morphometric analysis of geomorphic and speleologic maps, Kastning (1983) concluded that orientations of sinkholes, dry valleys and caves correspond to sets of fractures in the Edwards Plateau of central Texas, the Mississippi Plateau of western Kentucky and the Helderberg Plateau of east-central New York. Hubbard (1984) stated that many sinkholes are concentrated along fold and fault structures in the central and northern Valley and Ridge province, Virginia. He explained that the reasons for this distribution are: (1) Concentrations of joints and cleavage fractures increase the permeability of carbonate rock units along faults and folds. (2) Inclined carbonate strata are commonly bordered by aquitards or aquicludes, which channel surface and ground water to and along the carbonate rocks of geological structure. Southworth (1984) used remote sensing data to demonstrate the alignment of sinkholes (cenotes) and inlets (caletas) on strike with existing faults and fracture systems in Yucatan Peninsula, Mexico. Black (1984) found that sinkholes tend to form along fault trends and ‘‘earth cracks’’ in northern Lower Michigan. Sinkholes exhibit alignments that are parallel to fault zones, joint sets, or vertical-bedding planes in the carbonate rocks of the Lehigh Valley, eastern Pennsylvania (Meyers and Perlow 1984). Littlefield et al. (1984) demonstrated that lineament intersection areas have the highest sinkhole probability and areas with unfractured rock have the lowest sinkhole probability in west-central Florida. Lineament analysis is also used in the field of structure geology or geophysics. Lutz (1986) used azimuth analysis of point like features to study the orientations of large-scale crustal structures. Lutz and Gutmann (1995) modified this method by using kernel density estimation to improve its performance on heterogeneous point distributions. Fractal analysis is a useful tool to study fracture connectivity and model fracture flow (Fuller and Sharp 1992).

Faivre and Reiffsteck (1999) measured strain and stress from sinkhole distribution in Velebit mountain range, Dinarides, Croatia using Panozzo’s projection method. Faivre and Reiffsteck’s (1999) research reveals that sinkhole development is closely related to tectonic activity in some cases. Nearest neighbor analysis Many individuals have attempted to study the patterns among point the data in various natural systems before the 1950s. Clark and Evans (1954) and Thompson (1956) formulated a NNA method and it has been used in many research areas such as geography, ecology, and geomorphology after the 1950s. Williams (1972) demonstrated that the distributions of sinkholes in eight regions of New Guinea were near random or approached uniform by means of Clark and Evans’s index, Kemmerly (1982) used Clark and Evans’ (1954) nearest-neighbor index (NNI) in many selected quadrangles in Kentucky and Tennessee to show that sinkholes are either clustered or randomly distributed in different quadrangles. Drake and Ford (1972) analyzed growth patterns of two generations of sinkholes in the Mendip Hills, England. By comparing the mean distances of the first to the twelfth nearest neighbors between two generations of sinkholes, Drake and Ford (1972) concluded that the number of daughter sinkholes associated with each of the parents is consistent over space. Based on the NNA of the distances between new and old sinkholes, Hyatt et al. (1999) concluded that most new sinkholes cluster around new sinkholes instead of old sinkholes and locations of old sinkholes could not be used to predict the new sinkhole development in Albany-Dougherty Plain, Georgia. However, the authors tested the NNA between the new sinkholes and old sinkholes within the Albany-Dougherty Plain instead of the Flint River flood limit within which the new sinkholes were located.

Previous study of sinkhole distribution in Minnesota Ruhl (1995) studied the relation of fracture orientation to linear terrain features in the Prairie du Chien Karst in southeastern Minnesota. Fracture orientations were measured in ten exposure sites at quarries, road cuts, and karst outcrops. Directions trends of linear terrain features were identified from 1:80,000 aerial photographs. Fracture orientations measured from two out of ten sites in Ruhl’s (1995) study correlates to linear terrain trends. He stated that the fracture patterns are not uniform in Prairie du Chien Karst and fracture orientations showed similar or different statistically significant directions at eight out of ten sites.

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Magdalene and Alexander (1995) applied NNA to the Winona County sinkhole dataset and demonstrated that sinkholes in Winona County were clustered by comparing with six random datasets using test statistics by Skellam (1952) and Clark and Evans (1954). Magdalene (1995) also studied the direction to the nearest neighbor for the Winona sinkhole dataset but the result is inconclusive with respect to the hypothesis of preferred orientation due to structural control. Gao et al. (2001) used NNA to test sinkhole distribution in different topographic and geologic settings. The sinkholes in southeastern Minnesota were categorized into three karst groups: Cedar Valley Karst (Middle Devonian), Galena/Spillville Karst (Upper Ordovician/Middle Devonian), and Prairie du Chien Karst (Lower Ordovician). NNA was conducted for sinkholes in each karst group. The median distance to the nearest neighbor (DNN) is significantly smaller than the mean DNN of all the sinkhole groups. NNA results on other sinkhole data sets all showed a highly skewed distribution. All current NNA results demonstrate that sinkholes in Minnesota are not evenly distributed in this area i.e., they tend to be clustered. This result confirms and expands Magdalene and Alexander’s (1995) conclusions with regard to the entire Minnesota data set. Sinkholes in the Prairie du Chien Karst are spaced about three to five times farther apart than the sinkholes in the Cedar Valley and Galena/Spillville Karst. This implies that more isolated sinkholes occur in Prairie du Chien Karst. NNA on sinkhole distribution was also tested in different counties, over different bedrock units, and in different probability areas. All the sinkholes in the sinkhole plains (highest sinkhole probability area) of Fillmore County were selected for an extended NNA and compared with Poisson and lognormal distributions.

Fig. 1 Comparison of the distribution of DNN of sinkholes in the sinkhole plains of Fillmore County with lognormal distribution

The distribution of DNN is distinctly different from the Poisson distribution within the sinkhole plains of Fillmore County. As the Poisson process describes randomly distributed data, this implies that sinkholes within the sinkhole plains of Fillmore County are not randomly distributed. The Poisson process does not adequately model the sinkhole distribution of Minnesota and may not be applicable to predict sinkhole occurrences. A comparison of the distribution of DNN and lognormal distribution indicates that the distribution within the sinkhole plains of Fillmore County matches a lognormal distribution (Fig. 1). The mean and standard deviation of DNN was used to define boundaries for extended NNA and sinkhole probability modeling (Gao and Alexander 2003).

Refined NNA of sinkhole distribution in Minnesota Methodology Karst areas defined for NNA To use NNI to test the complete spatial randomness (csr) of sinkhole distribution in Minnesota, karst areas need to be defined and the analysis should be confined to sinkholes occurring in the defined areas. Previous studies of sinkhole distribution in Minnesota (Alexander and Maki 1988; Dalgleish 1985; Dalgleish and Alexander 1984; Green et al. 1997; Magdalene 1995; Tipping et al. 2001; Witthuhn and Alexander 1995) indicated that the primary controls of sinkhole distribution are bedrock geology and depth to bedrock. Magdalene (1995) used areas underlain by Prairie du Chien formation to calculate NNI for sinkholes in Winona County. The carbonate bedrock boundary can be reduced to contain only shallow carbonate bedrock. Gao et al. (2002) indicated that almost all sinkholes occur in the overlapped areas underlain by carbonate bedrock and areas where depth to bedrock is less than 15 m (50 feet). ‘‘50 feet’’ is the normal minimum contour interval on Minnesota’s standard ‘‘Depth to Bedrock Maps’’ — which are parts of all the County Geologic Atlases. As the dissolution of carbonate rocks occurs far from the surface, it is unlikely to see sinkholes in areas with more than 50 feet of sediment cover in Minnesota. This phenomenon also appears in Iowa. Hallberg and Hoyer (1982) stated that 12,700 mapped sinkholes in northeastern Iowa were only found in certain areas where the ‘‘soil materials’’ are less than 9 m (30 feet) thick. Table 1 lists a revised karst grouping in southeastern Minnesota. The Prairie du Chien Karst remains the same as that used by Gao et al. (2001). The Galena/ Spillville and Cedar Valley Karst in Gao et al. (2001) were revised as Galena–Maquoketa Karst and the Devonian Karst by:

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Table 1 Karst groups in southeastern Minnesota as used in spatial analysis and probability modeling Series

Group, Formation, Member

Unit Symbol

Karst Group

Middle Devonian

Lithograph City Formation – Hinkle & Eagle Center Mbrs Chickasaw Member – Spillville Formation

DCUU DCUM DCLC DCLP DSPL OMAQ ODUB OGAL

Devonian Karst

Upper Ordovician

Lower Ordovician

Maquoketa Formation – Cummingsville Formation

Decorah Shale – Glenwood Formation St. Peter Sandstone Prairie du Chien Group

a

Devonian Karst Galena–Maquoketa Karst OGSV OGPR OGCM

ODPG OSTP OPDC

OPSH OPOD

b

Prairie du Chien Karst

a The Devonian Karst as defined above may need to be subdivided in future work b For map compatibility reasons, the map bottom of the GalenaMaquoketa Karst in this paper is taken to be the Cummingsville/

Prosser (OGCM/OGPR) contact. The top two thirds of the Cummingsville Formation is an active part of this karst, but does not contain many sinkholes

1. Removing the areas covered by more than 50 feet of surficial cover. 2. Excluding the Cummingsville (OGCM) formation. Sinkholes are rarely mapped on the Cummingsville formation. However, the upper two-thirds of the Cummingsville formation hosts many large caves and is part of the Galena–Maquoketa Karst. 3. Moving the Spillville (DSPL) formation from the Galena/Spillville used by Gao et al. (2001) to the Devonian Karst. A major erosional surface exists between the Spillville and the Maquoketa (OMAQ) Formations. According to Hallberg and Hoyer (1982), the majority of the sinkholes in northeast Iowa are concentrated in three areas: one area underlain by Galena formation (OGAL), in southwestern Allamakee County and adjacent areas; two areas underlain by Silurian–Devonian carbonate rocks, in southern Clayton County and adjacent areas, and areas adjacent to the Cedar River in Floyd and Michell Counties. Figure 2 shows that the sinkhole distribution in DSPL in southeastern Minnesota follows the pattern of sinkhole distribution in Silurian–Devonian karst in northeastern Iowa. Patterns of sinkhole distribution between DSPL and OGAL groups are clearly separated in northeastern Iowa but merge in Fillmore County. At a regional scale including both southeastern Minnesota and northeastern Iowa, DSPL should be combined with Cedar Valley Karst to form Devonian Karst.

Minnesota, sinkhole datasets in Goodhue, Wabasha, Olmsted, Fillmore, and Mower Counties were selected for NNA and probability modeling. Reasonably complete datasets for bedrock geology, depth to bedrock, and sinkhole locations exist in these five counties. Other counties have not been as fully mapped for depth to bedrock or sinkhole locations. Figure 3 is the cartographic model flowchart to construct active karst areas in Goodhue, Wabasha, Olmsted, Fillmore, and Mower Counties. Active karst areas are defined as areas underlain by carbonate rocks and the depth to carbonate is less than 50 feet. Bedrock geology and depth to bedrock in each county are reclassified as karst groups and shallow (<50 feet) or deep bedrocks (>50 feet). Boundaries between the same karst groups were dissolved and the result was then converted into a karst area coverage for each county. Selected shallow bedrock polygons in each county were converted into shallow bedrock coverages. Shallow bedrock coverages from different counties were merged and the topography was cleaned up to build a shallow bedrock coverage for the five counties. Karst areas from different counties were then merged and the topography was cleaned up to build a karst area coverage for the five counties. The active karst coverage for Goodhue, Wabasha, Olmsted, Fillmore, and Mower Counties was formed by the intersection of the five county karst coverage and the shallow bedrock coverage. The three active karst areas in these five counties are used for NNA and sinkhole probability modeling. Two intermediate coverages of this model, karst areas and shallow bedrock, were also used to verify sinkhole locations and the boundaries of bedrock geology and depth to bedrock maps.

Cartographic modeling Based on available data sources for bedrock geology, depth to bedrock, and data stored in the KFD of

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Fig. 2 Sinkhole distribution and bedrock geology in southeastern Minnesota and northeastern Iowa. The majority of bedrock geology information in Minnesota was compiled from 1:100,000 scale maps, Minnesota Geological Survey (Table 2). Iowa bedrock geology information was compiled from 1:250,000 scale maps, Iowa Department of Natural Resources, Geological Survey Bureau (Witzke et al. 1998; 2001). Michael Bounk at the Iowa Geological Survey Bureau provided Iowa sinkhole dataset

Distances to the first and Nth nearest neighbor The Clark and Evans (1954) index R has been widely used to evaluate the csr for point feature distributions. Table 3 lists the symbols and definitions used to calculate the R and to test the csr for sinkhole distribution in southeastern Minnesota.

The values of R in Table 3 range between 0 and 2.1491 (Clark and Evans 1954). If R is closer to 1, the point pattern is random. If R < 1, the smaller the R, the more clustered the point pattern displays. If R > 1, the greater the R, the more regular the point pattern displays. In Table 3, c is used to test the statistical significance of R. For a two-tailed test, the |c| values of 1.96 and 2.58 correspond to the 0.05 and 0.01 levels of significance. Clark and Evans’ (1954) method applies to the first nearest neighbor only. Thompson (1956) formulated the distances to the Nth nearest neighbor. Definitions of mean distance to the Nth nearest neighbor expected from csr and its standard error are listed in Table 4.

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Table 2 Map and digital data sources for bedrock geology in southeastern Minnesota County or Multi-county area

Map Source

Fillmore Goodhue Houston Mower Olmsted Rice Wabasha Winona

Mossler (1995) Runkel (1998) Runkel (1996) Mossler (1998) Olsen (1988) Mossler (1995) Mossler (2001) Mossler and Book (1984) Mossler (2002)

MGS MGS MGS MGS MGS MGS MGS MGS

Steele, Dodge, Olmsted and western Winona Seven-county metropolitan Twin Cities area 13 counties of south central Minnesota

Table 3 Symbols and equations used to calculate and evaluate Clark and Evans (1954) index (Adapted from Clark and Evans 1954)

Digital Source Symbols

Definition

N

d

N/A

rE

0:5 p ffiffi d

MGS

rA R S(rE)

rA/rE 0:2614 pffiffiffiffi

Mossler and Tipping (2000)

MGS

c

Water Resources Center (1999)

Water Resources Center Mankato State University

Preliminary NNA

R and c calculated from the first to the fourth nearest neighbor distances are used to investigate different patterns of sinkhole distribution. A nearest-neighbor program written in C was used to conduct the analysis. The program uses the UTM coordinates of sinkhole locations to calculate the direction and distance to each sinkhole’s nearest neighbor. Several ArcView nearest neighbor analyst extensions (Colin 1998; Saraf 2002; Weigel 2002) were implemented on different sets of sinkhole data. All of these methods generate comparable results.

Fig. 3 Cartographic modeling flow chart to create active karst areas in Goodhue, Wabasha, Olmsted, Fillmore, and Mower Counties. The active karst areas are used for near neighbor analysis and sinkhole probability modeling

P

r

n

nd rA
rE rðrEÞ

Description Total number of sinkholes within the area where NNA is conducted Sinkhole density within the tested area (A is area) Mean DNN expected if the sinkhole distributed is in csr The actual mean DNN Clark and Evans (1954) index The standard error of rE The standard variate of the normal curve

A preliminary NNA of sinkhole distributions was conducted within the three active karst areas in Goodhue, Wabasha, Olmsted, Fillmore, and Mower Counties (Fig. 4). In an infinite population, sinkholes close to the boundary of a study area may have nearest-neighbors outside of the boundary. Limiting the analysis to sinkholes within the boundary may result in DNN values that are too high. There are hundreds of sinkholes close to the county or active karst boundaries within the five county areas. Some of those sinkholes may have a nearest neighbor that lies outside of the county or active boundaries. This phenomenon is called edge effect. Conducting Clark and Evans (1954) test without removing edge effects is likely to be biased towards

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Table 4 rE and r(r) for distances to the Nth nearest Neighbor (Adapted from Thompson 1956) N

1

2

3

4

rE

0:5 p ffiffi d 0:2614 pffiffiffiffi nd

0:75 pffiffi d 0:2723 pffiffiffiffi nd

0:9375 pffiffi d 0:2757 pffiffiffiffi nd

1:0937 pffiffi d 0:2774 pffiffiffiffi nd

S(rE)

Fig. 5 Two different point patterns not distinguishable by the distance to the first nearest neighbor

Fig. 4 Sinkhole Distribution and active karst areas in Goodhue, Wabasha, Olmsted, Fillmore, and Mower Counties

regular patterns. This biased result is more evident if the sample size is less than 100 (Krebs 1989). One way to eliminate edge effects is to add a boundary strip or buffer zone inside the border of the study area (Cressie 1993). DNN is only calculated for the points within the buffered area. Points within the boundary strip still count as potential neighbors for points falling inside the buffer zones. Another approach to remove edge effects is to eliminate points whose distances to the border of study area are less than their DNN. The second approach is more applicable for NNA of sinkhole distribution because the dendritic patterns of bedrock topography and bedrock geology make it difficult to define appropriate buffer zones. To avoid edge effects, sinkholes were evaluated for proximity to the county or active karst boundaries. Some isolated sinkholes are very far away from the main

populations of sinkholes. These areas have not been fully investigated and some sinkholes might exist but may not be mapped or recorded in the KFD. Three kinds of sinkholes were removed for NNA: sinkholes that have nearest neighbors outside of the study area whose DNN patterns are significantly different from those in the study area, sinkholes whose distances to their nearest neighbors are greater than the distance to the boundary of the study area, and some isolated sinkholes whose neighborhood has not been fully mapped for karst features. Table 5 shows the results of a preliminary NNA for sinkholes within the three active karst areas. Figure 4 and Table 5 demonstrate that sinkholes in the three active karst areas are strongly clustered in some small areas. Sinkholes in the Prairie du Chien Karst have the lowest density (0.15 km)2) and sinkholes in Galena– Maquoketa Karst have the highest density (4.82 km)2). Devonian Karst and Galena–Maquoketa Karst have highly concentrated areas containing hundreds of sinkholes. Comparing to sinkholes in Devonian Karst and Galena–Maquoketa Karst, isolated sinkholes occur more often in Prairie du Chien Karst. Extended NNA Using NNA or NNI alone can lead to misleading results for some distribution patterns. Getis (1963) used NNA and associated quadrat analysis to analyze and to test the evolution of land use patterns in light of population density and transportation changes. Chou (1997) also pointed out that spatial autocorrelation statistics using Moran’s I coefficient (Moran 1948) is a useful approach to study the spatial patterns.

Table 5 Preliminary NNA tests for sinkholes distributed in three karst areas at Goodhue, Wabasha, Olmsted, Fillmore, and Mower Counties Karst group

Area (km2)

n

d (no./km2)

rE (m)

rA(m)

R

r (rE)

c

Devonian Galena–Maquoketa Prairie du Chien

665 1380 3070

525 6657 455

0.79 4.82 0.15

562.73 227.65 1298.77

99.10 83.20 358.30

0.18 0.37 0.28

12.84 1.46 31.83

-36.11 -99.03 -29.54

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Table 6 NNA of distances of the first to the fourth nearest sinkholes n

d (no./km2)

rE (m)

rA(m)

R

r (rE)

c

Distances to the first nearest neighbor: Devonian 665 Galena–Maquoketa 1380 Prairie du Chien 3070

525 6657 455

0.79 4.82 0.15

562.73 227.65 1298.77

99.1 83.2 358.3

0.18 0.37 0.28

12.84 1.46 31.83

)36.11 )99.03 )29.54

Distances to the second nearest neighbor: Devonian 665 Galena–Maquoketa 1380 Prairie du Chien 3070

525 6657 455

0.79 4.82 0.15

844.1 341.48 1948.16

166 125.3 693.8

0.2 0.37 0.36

13.38 1.52 33.16

)50.7 )142.3 )37.83

Distances to the third nearest neighbor: Devonian 665 Galena–Maquoketa 1380 Prairie du Chien 3070

525 6657 455

0.79 4.82 0.15

1055.12 426.85 2435.2

225 160.5 1071

0.21 0.38 0.44

13.54 1.54 33.57

)61.3 )173.1 )40.63

Distances to the fourth nearest neighbor: Devonian 665 Galena–Maquoketa 1380 Prairie du Chien 3070

525 6657 455

0.79 4.82 0.15

123092 497.96 2840.94

287 194.5 1347

0.23 0.39 0.47

13.63 1.55 33.78

)69.28 )195.1 )44.23

Karst Group

Area (km2)

Instead of using supplemental analytical methods such as quadrat analysis and spatial autocorrelation, NNA was substantially extended to calculate distances up to the ninth nearest neighbor, sinkhole distribution in different zones within an active karst area, and directions to the nearest neighbor. Distances to the Nth nearest neighbor Nearest neighbor analysis, especially using the distances to the first nearest neighbor, may overlook other spatial relations that exist in the study area. Patterns of scattered small clusters cannot be differentiated with patterns of a large cluster if their DNN are identical. For example, Fig. 5 illustrates two completely different point patterns that are not distinguishable by using NNI for the distance to the first nearest neighbor because the two patterns have almost identical DNN. NNI should be extended to the distances measured from the second nearest neighbor, the third nearest neighbor, and so forth for complex patterns (Aplin 1983; Chou 1997). Table 6 presents the NNA results of distances to the first – fourth nearest sinkholes in the three active karst areas in Goodhue, Wabasha, Olmsted, Fillmore, and Mower Counties. The NNI, R, of different orders of NNA is very consistent in Devonian and Galena–Maquoketa Karsts. In Prairie du Chien Karst, the NNI increases from 0.28 for the first NNA to 0.47 for the fourth NNA. Comparisons of median distances to the Nth nearest sinkholes in different karst areas are shown in Fig. 6. Sinkholes in the Devonian and the Galena–Maquoketa Karsts are more clustered than the sinkholes in the Prairie du Chien Karst. The median distances to the ninth nearest sinkholes in Devonian and Galena–

Maquoketa Karsts are less than the median distance to the second nearest sinkholes in Prairie du Chien Karst. P–P plot, also called probability plot, is a graph that plots cumulative proportions against the cumulative proportions of any of a number of test distributions. Probability plots are generally used to determine whether the distribution of a variable matches a given distribution. If the selected variable matches a lognormal distribution, the points cluster around a straight line. Figures 7, 8, and 9 illustrate lognormal P–P plots for distances to the first – third nearest sinkholes in the active karst areas in Goodhue, Wabasha, Olmsted, Fillmore, and Mower Counties. In Devonian and Galena–Maquoketa Karst areas, distances to the first and second nearest sinkholes closely match the lognormal distribution. Distances to the third nearest sinkholes start to depart from the lognormal distribution. This trend becomes more evident for distances beyond the third nearest sinkholes (not shown). In Prairie du Chien Karst areas, distances to the first nearest sinkholes marginally match the lognormal distribution; distances beyond the first nearest sinkholes do not match the lognormal distribution. Sinkhole distribution in different zones in the Galena–Maquoketa Karst In NNA, the results can vary depending on the scale of the area analyzed. Sinkholes are not uniformly distributed in the active karst areas shown in Fig. 4. The NNA analysis was extended to areas of high sinkhole density within each active karst area. Figures 7, 8, 9 illustrate that the DNN of the majority of the sinkhole population matches lognormal distribution in southeastern Minnesota. Table 7 lists the mean and standard deviation of distances to the nearest sinkholes derived from lognor-

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Fig. 6 Median distances to the Nth nearest sinkholes in the active karst areas in Goodhue, Wabasha, Olmsted, Fillmore, and Mower Counties. a Devonian Karst. b Galena-Maquoketa Karst. c Prairie du Chien Karst

mal distribution for different karst groups. If the DNN matches lognormal distribution, DNN of 95% of the Galena–Maquoketa sinkholes is about 200 m; DNN of 95% of Devonian sinkholes and 99% of the Galena– Maquoketa sinkholes are about 400 m; DNN of 70% of Prairie du Chien sinkholes is about 700 m. Therefore, 200, 400, and 700 m were used to define concentrated sinkhole zones for NNA. Sinkholes are less than 200, 400, or 700 m to their nearest neighbor within these c

Fig. 7 Lognormal P–P plot for distances to the first to the third nearest sinkholes in Devonian Karst

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b

Fig. 8 Lognormal P–P plot for distances to the first to the third nearest sinkholes in Galena-Maquoketa Karst

zones. Concentrated zones with more than 100 sinkholes are used to calculate distances and directions to the nearest sinkholes. Figure 10 illustrates that most of the highly concentrated sinkholes are in the Galena–Maquoketa Karst in Fillmore County. Two small zones in the white areas fall in the Devonian Karst. One zone falls in the Galena– Maquoketa Karst in Olmsted County. The sinkholes within the concentrated zones shown in Fig. 10 represent more than 90% of all of the mapped sinkholes in the Minnesota karst dataset. Results of NNA in different concentrated zones in Fig. 10 are presented in Tables 8 and 9. The NNIs listed in Table 9 indicate that many sinkholes are concentrated in these zones. By changing the zone scale from 700 m to 200 m, the majority of the sinkhole distributions change from clustered to random or regular patterns. This trend is more evident in the nested zones, the smaller the zone scale, the higher the NNI, the more likelihood that sinkhole patterns change to more regular patterns. Directions to the nearest neighbor Magdalene’s (1995) study concluded that the distribution of directions to the nearest neighbor for the Winona county sinkhole dataset were not significantly different from the distribution derived from randomly distributed samples in the same setting. As a number of the sinkholes in highly concentrated zones display regular patterns, directions to the nearest sinkhole were calculated in different scales of zones to investigate the pattern change with the zone scale change. Figure 11 illustrates that directions to the nearest sinkholes in the Galena–Maquoketa do not show orientation preferences. Directions to the nearest sinkholes in the Prairie du Chien Karst present some orientation preferences of East–West and NW–SE. Directions to the nearest sinkholes in Devonian Karst present a major orientation close to East–West. This orientation preference is consistent in the two concentrated zones (with a minimum 700 m DNN) in Devonian Karst (Fig. 12). Nearest neighbor index calculated from DNN indicates that the majority of the sinkhole distributions change from clustered to random or regular patterns along with the zone scale changing from 700 m to 400 and 200 m. This trend is also revealed in the directions to the nearest sinkholes. Figure 13 illustrates rose diagram plots of directions to the nearest sinkholes in nested concentrated zones in Galena–Maquoketa karst. Each smaller concentrated zone is within a larger zone.

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b

Fig. 9 Lognormal P–P plot for distances to the first to the third nearest sinkholes in Prairie du Chien Karst

When the zone scale becomes smaller, the major orientation preference becomes more evident.

Results and discussion Clark and Evans’s (1954) test of sinkholes in the three active karst areas indicates that sinkholes tend to be clustered in each active karst area. Maps of sinkhole density and sinkhole distribution in southeastern Minnesota display some highly concentrated zones of sinkholes, especially in the active karst areas of Devonian and Galena–Maquoketa Karsts. These areas of concentrated sinkholes were delineated based on the mean and standard deviation that corresponds to the DNN’s lognormal distribution. Sinkholes are highly concentrated in Galena–Maquoketa and Devonian Karst. In Galena–Maquoketa Karst, approximately 30% sinkholes concentrate in only 2% of the total active karst area, the 200-m zones in Fig. 10. About 90% sinkholes concentrate in 10% of the total karst area, the 400-m zones in Fig. 10. In Devonian Karst, more than 70% sinkholes concentrate in only 3% of the total active karst area, the 700-m zones in Fig. 10. The scale effect on sinkhole distribution has been investigated in other karst areas in the U.S. Zhou et al. (2003) conducted orientation analysis of sinkholes along I-70 highway near Fredrick, Maryland. The analysis demonstrated that orientations of sinkhole pairs at different scales correspond to regional and local structures such as the axis of regional syncline, the strike of rock units, major joints, and secondary set of joints. Analyses of sinkhole distribution in different zones within an active karst area, and directions to the nearest neighbor indicate that sinkhole distributions change from clustered to random to regular when the minimum DNN of the concentrated zone decreases. Figure 14 illustrates that directions to the nearest sinkholes in many highly concentrated zones (200 m zone) have evident orientation preference, especially for zones within which sample size is less than 200. One hypothesis that can explain this pattern in southeastern Minnesota Table 7 Mean and standard deviation of distances to the nearest sinkholes derived from lognormal distribution Karst group

Mean rA(m)

rA + r

rA + 2r

rA + 3r

Devonian Galena–Maquoketa Prairie du Chien

67 69 134

168 126 655

417 233 3,215

1,036 429 155,771

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Fig. 10 Concentrated zones of sinkholes in southeastern Minnesota

Table 8 Distances to the nearest sinkholes in concentrated zones in Devonian, and Galena-Maquoketa Karst areas Devonian Karst: Minimum DNN (m)

Zone#

Area (km2)

N

d (no./km2)

R

c

Pattern

700 400 700 Galena-Maquoketa Karst: 200 200 200 200 200 200 200 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 700 700 700 700 700 700

226 1,288 119

9.23 2.05 12.2

188 110 119

20.37 53.66 9.75

0.59 0.74 0.82

)10.85 )5.23 )3.75

Cluster Cluster Cluster

1,217 1,273 1,409 1,576 1,968 1,947 1,935 640 850 868 886 891 959 996 998 1,036 1,108 1,138 1,183 1,189 1,193 1,258 85 104 42 134 158 175

10.31 1.74 2.86 2.18 7.55 1.64 5.31 8.71 3.77 16.7 22 2.72 4.44 5 7.56 3.62 8.56 8.03 28.4 11.8 2.59 7.51 99.9 4.5 12.2 13.2 19.9 85.8

706 130 174 221 461 125 355 295 111 791 724 163 145 128 347 140 186 250 948 467 126 220 2175 172 296 203 256 1935

68.48 74.71 60.84 101.38 61.06 76.22 66.85 33.87 29.44 47.37 32.91 59.93 32.66 25.60 45.90 38.67 21.73 31.13 33.38 39.58 48.65 29.29 21.77 38.22 24.26 15.38 12.86 22.55

1.18 1.08 1.25 1.13 1.19 1.16 1.14 0.90 1.00 1.02 0.92 0.66 0.88 1.01 0.97 0.82 0.90 0.89 0.95 0.93 0.95 0.89 0.86 0.52 0.78 0.77 0.71 0.85

9.38 1.78 6.29 3.80 7.82 3.45 4.97 )3.13 )0.09 1.30 )4.20 )8.37 )2.69 0.19 )1.07 )4.15 )2.51 )3.24 )3.08 )2.88 )1.08 )3.19 )12.74 )12.10 )7.08 )6.35 )8.84 )13.04

Regular Random Regular Regular Regular Regular Regular Cluster Random Random Cluster Cluster Cluster Random Random Cluster Cluster Cluster Cluster Cluster Random Cluster Cluster Cluster Cluster Cluster Cluster Cluster

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Table 9 Distances to the nearest sinkholes in the nested zones in Glena–Maquoketa Karst Minimum DNN (m)

Zone#

Area (km2)

N

d (no./km2)

R

c

Pattern

700 400 700 400 700 400 700 400 700 400 400 200 400 200 200 400 200 400 400 700 400 200 400 200 400 200 400 400

104 891 42 640 134 996 158 1,108 85 850 868 1,217 886 1,273 1,409 998 1,576 959 1,036 175 1,183 1,968 1,189 1,935 1,193 1,947 1,138 1,258

4.5 2.72 12.2 8.71 13.2 5 19.9 8.56 99.9 3.77 16.7 10.31 22 1.74 2.86 7.56 2.18 4.44 3.62 85.8 28.4 7.55 11.8 5.31 2.59 1.64 8.03 7.51

172 163 296 295 203 128 256 186 2175 111 734 665 640 108 162 236 169 111 94 1935 876 430 429 327 110 109 240 195

38.22 59.93 24.26 33.87 15.38 25.60 12.86 21.73 21.77 29.44 43.95 64.50 29.09 62.07 56.64 31.22 77.52 25.00 25.97 22.55 30.85 56.95 36.36 61.58 42.47 66.46 29.89 25.97

0.52 0.66 0.78 0.90 0.77 1.01 0.71 0.90 0.86 1.00 1.08 1.27 0.97 1.21 1.29 1.04 1.20 0.94 0.91 0.85 0.99 1.25 0.98 1.19 1.06 1.30 0.93 0.94

)12.10 )8.37 )7.08 )3.13 )6.35 0.19 )8.84 )2.51 )12.74 )0.09 4.26 13.17 )1.24 4.20 7.15 1.18 5.00 )1.23 )1.76 )13.04 )0.30 9.84 )0.76 6.74 1.16 5.92 )2.20 )1.54

Cluster Cluster Cluster Cluster Cluster Random Cluster Cluster Cluster Random Regular Regular Random Random Regular Random Regular Random Random Cluster Random Regular Random Regular Random Regular Cluster Random

is that structural patterns in these zones favor sinkhole formation. Structural patterns such as the densities and orientations of fractures, joints, and conduit systems should be systematically investigated in these areas. Lineament analysis of sinkhole distribution and comparison between structural pattern and sinkhole distribution need to be conducted in the near future to reach a conclusive explanation of sinkhole distribution in southeastern Minnesota. Nearest neighbor analysis and sinkhole density distribution in southeastern Minnesota illustrate that some highly concentrated sinkholes are in the middle of less concentrated sinkholes. In many concentrated zones, sinkhole density decreases gradually from the center to

Fig. 12 Rose diagram plots of directions to the nearest sinkholes in two concentrated zones in Devonian Karst (see Table 8 for results of NNA)

Fig. 11 Rose diagram plots of directions to the nearest sinkholes in the active karst areas (GEOrient 9.0 (Holcombe 2001) was used to plot rose diagrams)

Fig. 13 Rose diagram plots of directions to the nearest sinkholes in nested concentrated zones in Galena-Maquoketa Karst (see Table 8 for results of NNA)

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Fig. 14 Rose diagram plots of directions to the nearest sinkholes in seven highly concentrated zones in Galena– Maquoketa Karst at Fillmore County

the edge of the zone. Many karst scientists explained this phenomenon as the tendency that new sinkholes form close to old sinkholes. This tendency can be explained by two processes: (1) areas in the highly concentrated zones of sinkholes have similar geologic and topographical settings that favor sinkhole formation; (2) existing sinkholes change the hydraulic gradient in the surrounding area and increase the solution and erosional processes that eventually form more new sinkholes. To test the hypotheses of sinkhole formation in southeastern Minnesota, hydraulic patterns need to be monitored and investigated in the concentrated areas of sinkholes. A simple and more realistic approach is to investigate the size of the sinkholes in those areas. If the second hypothesis is true, larger and older sinkholes should form near the center of the concentrated zone and smaller and younger sinkholes should form near the edge of the concentrated zone.

Conclusions The distributions of distances to the first through third nearest sinkholes match lognormal distributions for the majority of the sinkhole population. Different concentrated zones of sinkholes in Devonian, Galena–Maquoketa Karst areas in Goodhue, Wabasha, Olmsted,

Fillmore, and Mower counties were delineated based on the lognormal distribution of the distances to the nearest sinkholes. NNA was extended to include different orders of NNA, different scales of concentrated zones, and directions to nearest sinkholes. The statistical results, along with the sinkhole density distribution, indicate that sinkholes tend to form in highly concentrated zones instead of scattered individuals. The pattern changes from clustered to random to regular as the scale of the analysis decreases from 10 – 100 km2 to 5 – 30 km2 to 2 – 10 km2. Hypotheses that may explain this phenomenon are: (1) areas in the highly concentrated zones of sinkholes have similar geologic and topographical settings that favor sinkhole formation; (2) existing sinkholes change the hydraulic gradient in the surrounding area and increase the solution and erosional processes that eventually form more new sinkholes. The following work would refine our understanding of sinkhole formation in southeastern Minnesota: 1. Investigate the size and orientation of the sinkholes in concentrated zones of sinkholes. 2. Monitor and investigate hydraulic patterns in the concentrated areas of sinkholes. 3. Systematically study the structural patterns such as joints and fractures and compare the lineament between structural patterns with the lineament of sinkhole distributions.

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