kfd implementation

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Introduction

This paper is the second of a series of two papersdescribing the Karst Feature Database (KFD) of Min-nesota (Gao et al. 2005 ). The development, implemen-tation, and data analyses of the Minnesota KFD aredescribed in detail in the rst authors Ph.D. dissertation(Gao 2002 ). This paper discusses the analysis of sinkholedistribution based on the sinkhole data stored in theMinnesota KFD.

The fundamental scientic question of this study iswhat controls sinkhole distribution in Minnesota. Acomplete statewide KFD provides data and tools to testhypotheses about sinkhole distribution in Minnesota.

Overview of spatial analysis on cave and karst studies

Karst scientists and researchers have developed and usedseveral analytic methods in cave and karst studies.Lineament analysis and nearest-neighbor analysis(NNA) are two of the most commonly used approachesto study karst feature distributions.

Lineament analysis

Geological structures in karst terrain are often ob-servable as traces or lines, commonly referred to aslineaments. Mapping and interpretation of lineaments

Y. GaoE. C. Alexander JrR. J. Barnes

Karst database implementation in Minnesota:analysis of sinkhole distribution

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

Abstract This paper presents theoverall sinkhole distributions andconducts hypothesis tests of sinkholedistributions and sinkhole formationusing data stored in the Karst Fea-ture Database (KFD) of Minnesota.Nearest neighbor analysis (NNA)was extended to include differentorders of NNA, different scales of concentrated zones of sinkholes, anddirections to the nearest sinkholes.The statistical results, along with thesinkhole density distribution, indi-cate that sinkholes tend to form inhighly concentrated zones instead of scattered individuals. The patternchanges from clustered to random toregular as the scale of the analysisdecreases from 10100 km 2 to530 km 2 to 210 km 2 . Hypothesesthat may explain this phenomenon

are: (1) areas in the highly concen-trated zones of sinkholes havesimilar geologic and topographicalsettings that favor sinkhole forma-tion; (2) existing sinkholes changethe hydraulic gradient in the sur-rounding area and increase thesolution and erosional processes thateventually form more new sinkholes.

Keywords Karst Feature database(KFD) Nearest neighbor analysis(NNA) Nearest neighbor index(NNI) Complete spatial random-ness (csr) Distance to nearestneighbor (DNN) Minnesota

Environ Geol (2005) 47: 10831098DOI 10.1007/s00254-005-1241-2 ORIGINAL ARTICLE

Y. Gao ( & )Department of Physics, Astronomy,and Geology, East Tennessee StateUniversity, Johnson City, TN 37604, USAE-mail: [email protected].: +1-423-4394183

E. Alexander JrDepartment of Geology & Geophysics,University of Minnesota,Minneapolis, MN 55455, USAE-mail: [email protected]

R. BarnesDepartment of Civil Engineering,University of Minnesota,Minneapolis, MN 55455, USAE-mail: [email protected]


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in relation to sinkhole distribution have been used inmany karst areas.

Matschinskis ( 1968 ) procedure was the rst todetermine the local alignment common to every threepoints (e.g. sinkholes) and then to plot the distributionof these local alignment directions on a histogram. Heconcluded from the interpretation of the histogram that

local alignments of sinkholes are mainly controlled bytectonics of the adjacent regions in Lake Constance,Germany.

Barlow and Ogden ( 1982 ) used a modied version of the KolmogorovSmirnov (KS) test to compare joints,straight-cave segments, and photo-lineament orienta-tions. They concluded that for low altitude photo-linea-ments and straight cave-passage segments, theorientations of joints, straight-cave segments, and photo-lineaments are similar in Benton County, Arkansas.Using evidence from eld work, interpretation of aerialphotography, and morphometric analysis of geomorphicand speleologic maps, Kastning ( 1983 ) concluded that

orientations of sinkholes, dry valleys and caves corre-spond to sets of fractures in the Edwards Plateau of cen-tral Texas, the Mississippi Plateau of western Kentuckyand the Helderberg Plateau of east-central New York.

Hubbard ( 1984 ) stated that many sinkholes are con-centrated along fold and fault structures in the centraland northern Valley and Ridge province, Virginia. Heexplained that the reasons for this distribution are: (1)Concentrations of joints and cleavage fractures increasethe permeability of carbonate rock units along faultsand folds. (2) Inclined carbonate strata are commonlybordered by aquitards or aquicludes, which channelsurface and ground water to and along the carbonate

rocks of geological structure.Southworth ( 1984 ) used remote sensing data todemonstrate the alignment of sinkholes (cenotes) andinlets (caletas) on strike with existing faults and fracturesystems in Yucatan Peninsula, Mexico. Black ( 1984 )found that sinkholes tend to form along fault trends andearth cracks in northern Lower Michigan. Sinkholesexhibit alignments that are parallel to fault zones, jointsets, or vertical-bedding planes in the carbonate rocks of the Lehigh Valley, eastern Pennsylvania (Meyers andPerlow 1984 ). Littleeld et al. ( 1984 ) demonstrated thatlineament intersection areas have the highest sinkholeprobability and areas with unfractured rock have thelowest sinkhole probability in west-central Florida.

Lineament analysis is also used in the eld of struc-ture geology or geophysics. Lutz ( 1986 ) used azimuthanalysis of point like features to study the orientationsof large-scale crustal structures. Lutz and Gutmann(1995 ) modied this method by using kernel densityestimation to improve its performance on heterogeneouspoint distributions. Fractal analysis is a useful tool tostudy fracture connectivity and model fracture ow(Fuller and Sharp 1992 ).

Faivre and Reiffsteck ( 1999 ) measured strain andstress from sinkhole distribution in Velebit mountainrange, Dinarides, Croatia using Panozzos projectionmethod. Faivre and Reiffstecks ( 1999 ) research revealsthat sinkhole development is closely related to tectonicactivity in some cases.

Nearest neighbor analysis

Many individuals have attempted to study the patternsamong point the data in various natural systems beforethe 1950s. Clark and Evans ( 1954 ) and Thompson(1956 ) formulated a NNA method and it has been usedin many research areas such as geography, ecology, andgeomorphology after the 1950s.

Williams ( 1972 ) demonstrated that the distributionsof sinkholes in eight regions of New Guinea were nearrandom or approached uniform by means of Clark andEvanss index, Kemmerly ( 1982 ) used Clark and Evans

(1954 ) nearest-neighbor index (NNI) in many selectedquadrangles in Kentucky and Tennessee to show thatsinkholes are either clustered or randomly distributed indifferent quadrangles.

Drake and Ford ( 1972 ) analyzed growth patterns of two generations of sinkholes in the Mendip Hills, Eng-land. By comparing the mean distances of the rst to thetwelfth nearest neighbors between two generations of sinkholes, Drake and Ford ( 1972 ) concluded that thenumber of daughter sinkholes associated with each of the parents is consistent over space. Based on the NNAof the distances between new and old sinkholes, Hyattet al. ( 1999 ) concluded that most new sinkholes cluster

around new sinkholes instead of old sinkholes andlocations of old sinkholes could not be used to predictthe new sinkhole development in Albany-DoughertyPlain, Georgia. However, the authors tested the NNAbetween the new sinkholes and old sinkholes within theAlbany-Dougherty Plain instead of the Flint River oodlimit within which the new sinkholes were located.

Previous study of sinkhole distribution in Minnesota

Ruhl ( 1995 ) studied the relation of fracture orientationto linear terrain features in the Prairie du Chien Karst insoutheastern Minnesota. Fracture orientations weremeasured in ten exposure sites at quarries, road cuts,and karst outcrops. Directions trends of linear terrainfeatures were identied from 1:80,000 aerial photo-graphs. Fracture orientations measured from two out of ten sites in Ruhls ( 1995 ) study correlates to linear ter-rain trends. He stated that the fracture patterns are notuniform in Prairie du Chien Karst and fracture orien-tations showed similar or different statistically signi-cant directions at eight out of ten sites.

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Magdalene and Alexander ( 1995 ) applied NNA tothe Winona County sinkhole dataset and demonstratedthat sinkholes in Winona County were clustered bycomparing with six random datasets using test statisticsby Skellam ( 1952 ) and Clark and Evans ( 1954 ). Mag-dalene ( 1995 ) also studied the direction to the nearestneighbor for the Winona sinkhole dataset but the result

is inconclusive with respect to the hypothesis of pre-ferred orientation due to structural control.Gao et al. ( 2001 ) used NNA to test sinkhole distri-

bution in different topographic and geologic settings.The sinkholes in southeastern Minnesota were catego-rized into three karst groups: Cedar Valley Karst(Middle Devonian), Galena/Spillville Karst (Upper Or-dovician/Middle Devonian), and Prairie du Chien Karst(Lower Ordovician). NNA was conducted for sinkholesin each karst group. The median distance to the nearestneighbor (DNN) is signicantly smaller than the meanDNN of all the sinkhole groups. NNA results on othersinkhole data sets all showed a highly skewed distribu-

tion. All current NNA results demonstrate that sink-holes in Minnesota are not evenly distributed in this areai.e., they tend to be clustered. This result conrms andexpands Magdalene and Alexanders ( 1995 ) conclusionswith regard to the entire Minnesota data set. Sinkholesin the Prairie du Chien Karst are spaced about three tove times farther apart than the sinkholes in the CedarValley and Galena/Spillville Karst. This implies thatmore isolated sinkholes occur in Prairie du Chien Karst.NNA on sinkhole distribution was also tested in differ-ent counties, over different bedrock units, and in dif-ferent probability areas. All the sinkholes in the sinkholeplains (highest sinkhole probability area) of Fillmore

County were selected for an extended NNA and com-pared with Poisson and lognormal distributions.

The distribution of DNN is distinctly different fromthe Poisson distribution within the sinkhole plains of Fillmore County. As the Poisson process describes ran-domly distributed data, this implies that sinkholeswithin the sinkhole plains of Fillmore County are notrandomly distributed. The Poisson process does notadequately model the sinkhole distribution of Minne-

sota and may not be applicable to predict sinkholeoccurrences. A comparison of the distribution of DNNand lognormal distribution indicates that the distribu-tion within the sinkhole plains of Fillmore Countymatches a lognormal distribution (Fig. 1). The meanand standard deviation of DNN was used to deneboundaries for extended NNA and sinkhole probabilitymodeling (Gao and Alexander 2003 ).

Rened NNA of sinkhole distribution in Minnesota

Methodology

Karst areas dened for NNA

To use NNI to test the complete spatial randomness(csr ) of sinkhole distribution in Minnesota, karst areasneed to be dened and the analysis should be conned tosinkholes occurring in the dened areas. Previous studiesof sinkhole distribution in Minnesota (Alexander andMaki 1988 ; Dalgleish 1985 ; Dalgleish and Alexander1984 ; Green et al. 1997 ; Magdalene 1995 ; Tipping et al.2001 ; Witthuhn and Alexander 1995 ) indicated that theprimary controls of sinkhole distribution are bedrockgeology and depth to bedrock. Magdalene ( 1995 ) used

areas underlain by Prairie du Chien formation to cal-culate NNI for sinkholes in Winona County. The car-bonate bedrock boundary can be reduced to containonly shallow carbonate bedrock. Gao et al. ( 2002 )indicated that almost all sinkholes occur in the over-lapped areas underlain by carbonate bedrock and areaswhere depth to bedrock is less than 15 m (50 feet).50 feet is the normal minimum contour interval onMinnesotas standard Depth to Bedrock Maps which are parts of all the County Geologic Atlases. Asthe dissolution of carbonate rocks occurs far from thesurface, it is unlikely to see sinkholes in areas with morethan 50 feet of sediment cover in Minnesota. This phe-nomenon also appears in Iowa. Hallberg and Hoyer(1982 ) stated that 12,700 mapped sinkholes in north-eastern Iowa were only found in certain areas where thesoil materials are less than 9 m (30 feet) thick.

Table 1 lists a revised karst grouping in southeasternMinnesota. The Prairie du Chien Karst remains thesame as that used by Gao et al. ( 2001 ). The Galena/Spillville and Cedar Valley Karst in Gao et al. ( 2001 )were revised as GalenaMaquoketa Karst and theDevonian Karst by:

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

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Distances to the rst and Nth nearest neighbor

The Clark and Evans ( 1954 ) index R has been widelyused to evaluate the csr for point feature distributions.Table 3 lists the symbols and denitions used to calcu-late the R and to test the csr for sinkhole distribution insoutheastern Minnesota.

The values of R in Table 3 range between 0 and2.1491 (Clark and Evans 1954 ). If R is closer to 1, thepoint 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 patterndisplays. In Table 3, c is used to test the statistical sig-nicance of R . For a two-tailed test, the | c| values of 1.96and 2.58 correspond to the 0.05 and 0.01 levels of sig-nicance.

Clark and Evans ( 1954 ) method applies to the rstnearest neighbor only. Thompson ( 1956 ) formulated thedistances to the N th nearest neighbor. Denitions of mean distance to the N th nearest neighbor expectedfrom csr and its standard error are listed in Table 4.

Fig. 2 Sinkhole distribution and bedrock geology in southeasternMinnesota and northeastern Iowa. The majority of bedrockgeology information in Minnesota was compiled from 1:100,000scale maps, Minnesota Geological Survey (Table 2). Iowa bedrockgeology 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 GeologicalSurvey Bureau provided Iowa sinkhole dataset

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R and c calculated from the rst to the fourth nearestneighbor distances are used to investigate differentpatterns of sinkhole distribution.

A nearest-neighbor program written in C was used toconduct the analysis. The program uses the UTMcoordinates of sinkhole locations to calculate the direc-tion and distance to each sinkholes nearest neighbor.Several ArcView nearest neighbor analyst extensions(Colin 1998 ; Saraf 2002 ; Weigel 2002 ) were implementedon different sets of sinkhole data. All of these methods

generate comparable results.

Preliminary NNA

A preliminary NNA of sinkhole distributions was con-ducted within the three active karst areas in Goodhue,Wabasha, Olmsted, Fillmore, and Mower Counties(Fig. 4). In an innite population, sinkholes close to theboundary of a study area may have nearest-neighborsoutside of the boundary. Limiting the analysis to sink-holes within the boundary may result in DNN valuesthat are too high. There are hundreds of sinkholes closeto the county or active karst boundaries within the vecounty areas. Some of those sinkholes may have anearest neighbor that lies outside of the county or activeboundaries. This phenomenon is called edge effect.Conducting Clark and Evans ( 1954 ) test without

removing edge effects is likely to be biased towards

Table 2 Map and digital data sources for bedrock geology insoutheastern Minnesota

County orMulti-county area

Map Source Digital Source

Fillmore Mossler ( 1995 ) MGSGoodhue Runkel ( 1998 ) MGSHouston Runkel ( 1996 ) MGSMower Mossler ( 1998 ) MGSOlmsted Olsen ( 1988 ) MGSRice Mossler ( 1995 ) MGSWabasha Mossler ( 2001 ) MGSWinona Mossler and Book

(1984 )MGS

Steele, Dodge,Olmsted andwestern Winona

Mossler ( 2002 ) MGS

Seven-countymetropolitanTwin Cities area

Mossler and Tipping(2000 )

MGS

13 counties of southcentral Minnesota

Water ResourcesCenter ( 1999 )

Water ResourcesCenter

Mankato StateUniversity

Table 3 Symbols and equations used to calculate and evaluateClark and Evans ( 1954 ) index (Adapted from Clark and Evans1954 )

Symbols Denition Description

N Total number of sinkholes withinthe area whereNNA is conducted

d N/A Sinkhole density withinthe tested area (A is area)

rE 0:5 ffiffi d p Mean DNN expected if thesinkhole distributed is in csrrA P

r

n The actual mean DNNR r A /rE Clark and Evans ( 1954 ) indexS (rE ) 0:2614 ffiffiffiffi nd p The standard error of rEc rA rE r rE The standard variate of thenormal curve

Fig. 3 Cartographic modelingow chart to create active karstareas in Goodhue, Wabasha,Olmsted, Fillmore, and MowerCounties. The active karst areasare used for near neighboranalysis and sinkhole probabil-ity modeling

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regular patterns. This biased result is more evident if thesample size is less than 100 (Krebs 1989 ).

One way to eliminate edge effects is to add aboundary strip or buffer zone inside the border of thestudy area (Cressie 1993 ). DNN is only calculated forthe points within the buffered area. Points within theboundary strip still count as potential neighbors forpoints falling inside the buffer zones. Another ap-proach to remove edge effects is to eliminate pointswhose distances to the border of study area are lessthan their DNN. The second approach is moreapplicable for NNA of sinkhole distribution becausethe dendritic patterns of bedrock topography andbedrock geology make it difficult to dene appropriatebuffer zones.

To avoid edge effects, sinkholes were evaluated forproximity to the county or active karst boundaries.Some isolated sinkholes are very far away from the main

populations of sinkholes. These areas have not beenfully investigated and some sinkholes might exist butmay not be mapped or recorded in the KFD. Threekinds of sinkholes were removed for NNA: sinkholesthat have nearest neighbors outside of the study areawhose DNN patterns are signicantly different fromthose in the study area, sinkholes whose distances totheir nearest neighbors are greater than the distance tothe boundary of the study area, and some isolatedsinkholes whose neighborhood has not been fully map-ped for karst features.

Table 5 shows the results of a preliminary NNA forsinkholes within the three active karst areas. Figure 4and Table 5 demonstrate that sinkholes in the threeactive karst areas are strongly clustered in some smallareas. Sinkholes in the Prairie du Chien Karst have thelowest density (0.15 km

) 2 ) and sinkholes in Galena Maquoketa Karst have the highest density (4.82 km

) 2 ).Devonian Karst and GalenaMaquoketa Karst havehighly concentrated areas containing hundreds of sink-holes. Comparing to sinkholes in Devonian Karst andGalenaMaquoketa Karst, isolated sinkholes occurmore often in Prairie du Chien Karst.

Extended NNA

Using NNA or NNI alone can lead to misleading resultsfor some distribution patterns. Getis ( 1963 ) used NNAand associated quadrat analysis to analyze and to testthe evolution of land use patterns in light of populationdensity and transportation changes. Chou ( 1997 ) alsopointed out that spatial autocorrelation statistics usingMorans I coefficient (Moran 1948 ) is a useful approachto study the spatial patterns.

Table 5 Preliminary NNA tests for sinkholes distributed in three karst areas at Goodhue, Wabasha, Olmsted, Fillmore, and MowerCounties

Karst group Area (km 2 ) n d (no./km 2 ) rE (m) rA (m) R r (rE ) c

Devonian 665 525 0.79 562.73 99.10 0.18 12.84 -36.11GalenaMaquoketa 1380 6657 4.82 227.65 83.20 0.37 1.46 -99.03Prairie du Chien 3070 455 0.15 1298.77 358.30 0.28 31.83 -29.54

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 ffiffi d p 0:75 ffiffi d p

0:9375 ffiffi d p

1:0937 ffiffi d p

S (rE ) 0:2614 ffiffiffiffi nd p 0:2723

ffiffiffiffi nd p 0:2757

ffiffiffiffi nd p 0:2774

ffiffiffiffi nd p

Fig. 5 Two different point patterns not distinguishable by thedistance to the rst nearest neighbor

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

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Instead of using supplemental analytical methodssuch as quadrat analysis and spatial autocorrelation,NNA was substantially extended to calculate distancesup to the ninth nearest neighbor, sinkhole distribution indifferent zones within an active karst area, and direc-tions to the nearest neighbor.

Distances to the Nth nearest neighbor

Nearest neighbor analysis, especially using the distancesto the rst nearest neighbor, may overlook other spatialrelations that exist in the study area. Patterns of scat-

tered small clusters cannot be differentiated with pat-terns of a large cluster if their DNN are identical. Forexample, Fig. 5 illustrates two completely different pointpatterns that are not distinguishable by using NNI forthe distance to the rst nearest neighbor because the twopatterns have almost identical DNN. NNI should beextended to the distances measured from the secondnearest neighbor, the third nearest neighbor, and soforth for complex patterns (Aplin 1983 ; Chou 1997 ).

Table 6 presents the NNA results of distances to therst fourth nearest sinkholes in the three active karstareas in Goodhue, Wabasha, Olmsted, Fillmore, andMower Counties. The NNI, R , of different orders of

NNA is very consistent in Devonian and GalenaMa-quoketa Karsts. In Prairie du Chien Karst, the NNIincreases from 0.28 for the rst NNA to 0.47 for thefourth NNA.

Comparisons of median distances to the N th nearestsinkholes in different karst areas are shown in Fig. 6.Sinkholes in the Devonian and the GalenaMaquoketaKarsts are more clustered than the sinkholes in thePrairie du Chien Karst. The median distances to theninth nearest sinkholes in Devonian and Galena

Maquoketa Karsts are less than the median distance tothe second nearest sinkholes in Prairie du Chien Karst.

P P plot, also called probability plot, is a graph thatplots cumulative proportions against the cumulativeproportions of any of a number of test distributions.Probability plots are generally used to determine whe-ther the distribution of a variable matches a given dis-tribution. If the selected variable matches a lognormaldistribution, the points cluster around a straight line.

Figures 7, 8, and 9 illustrate lognormal P P plots fordistances to the rst third nearest sinkholes in theactive karst areas in Goodhue, Wabasha, Olmsted,Fillmore, and Mower Counties. In Devonian and Ga-

lenaMaquoketa Karst areas, distances to the rst andsecond nearest sinkholes closely match the lognormaldistribution. Distances to the third nearest sinkholesstart to depart from the lognormal distribution. Thistrend becomes more evident for distances beyond thethird nearest sinkholes (not shown). In Prairie du ChienKarst areas, distances to the rst nearest sinkholesmarginally match the lognormal distribution; distancesbeyond the rst nearest sinkholes do not match thelognormal distribution.

Sinkhole distribution in different zonesin the GalenaMaquoketa Karst

In NNA, the results can vary depending on the scale of the area analyzed. Sinkholes are not uniformly distrib-uted in the active karst areas shown in Fig. 4. The NNAanalysis was extended to areas of high sinkhole densitywithin each active karst area. Figures 7, 8, 9 illustratethat the DNN of the majority of the sinkhole populationmatches lognormal distribution in southeastern Minne-sota. Table 7 lists the mean and standard deviation of distances to the nearest sinkholes derived from lognor-

Table 6 NNA of distances of the rst to the fourth nearest sinkholes

Karst Group Area (km 2 ) n d (no./km 2 ) rE (m) rA (m) R r (rE ) c

Distances to the rst nearest neighbor:Devonian 665 525 0.79 562.73 99.1 0.18 12.84 ) 36.11GalenaMaquoketa 1380 6657 4.82 227.65 83.2 0.37 1.46 ) 99.03Prairie du Chien 3070 455 0.15 1298.77 358.3 0.28 31.83 ) 29.54

Distances to the second nearest neighbor:Devonian 665 525 0.79 844.1 166 0.2 13.38 ) 50.7GalenaMaquoketa 1380 6657 4.82 341.48 125.3 0.37 1.52 ) 142.3Prairie du Chien 3070 455 0.15 1948.16 693.8 0.36 33.16 ) 37.83

Distances to the third nearest neighbor:Devonian 665 525 0.79 1055.12 225 0.21 13.54 ) 61.3GalenaMaquoketa 1380 6657 4.82 426.85 160.5 0.38 1.54 ) 173.1Prairie du Chien 3070 455 0.15 2435.2 1071 0.44 33.57 ) 40.63

Distances to the fourth nearest neighbor:Devonian 665 525 0.79 123092 287 0.23 13.63 ) 69.28GalenaMaquoketa 1380 6657 4.82 497.96 194.5 0.39 1.55 ) 195.1Prairie du Chien 3070 455 0.15 2840.94 1347 0.47 33.78 ) 44.23

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mal distribution for different karst groups. If the DNNmatches lognormal distribution, DNN of 95% of theGalenaMaquoketa 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 dene concentratedsinkhole zones for NNA. Sinkholes are less than 200,400, or 700 m to their nearest neighbor within these

Fig. 7 Lognormal P P plot for distances to the rst to the thirdnearest sinkholes in Devonian Karst

c

Fig. 6 Median distances to the N th nearest sinkholes in the activekarst areas in Goodhue, Wabasha, Olmsted, Fillmore, and MowerCounties. a Devonian Karst. b Galena-Maquoketa Karst. c Prairiedu Chien Karst

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zones. Concentrated zones with more than 100 sinkholesare used to calculate distances and directions to thenearest sinkholes.

Figure 10 illustrates that most of the highly concen-trated sinkholes are in the GalenaMaquoketa Karst inFillmore County. Two small zones in the white areas fallin the Devonian Karst. One zone falls in the Galena Maquoketa Karst in Olmsted County. The sinkholeswithin the concentrated zones shown in Fig. 10 repre-sent more than 90% of all of the mapped sinkholes inthe Minnesota karst dataset. Results of NNA in differ-ent concentrated zones in Fig. 10 are presented inTables 8 and 9.

The NNIs listed in Table 9 indicate that many sink-holes are concentrated in these zones. By changing thezone scale from 700 m to 200 m, the majority of the

sinkhole distributions change from clustered to randomor regular patterns. This trend is more evident in thenested zones, the smaller the zone scale, the higher theNNI, the more likelihood that sinkhole patterns changeto more regular patterns.

Directions to the nearest neighbor

Magdalenes ( 1995 ) study concluded that the distribu-tion of directions to the nearest neighbor for the Winonacounty sinkhole dataset were not signicantly differentfrom the distribution derived from randomly distributedsamples in the same setting. As a number of the sink-

holes in highly concentrated zones display regular pat-terns, directions to the nearest sinkhole were calculatedin different scales of zones to investigate the patternchange with the zone scale change.

Figure 11 illustrates that directions to the nearestsinkholes in the GalenaMaquoketa do not show ori-entation preferences. Directions to the nearest sink-holes in the Prairie du Chien Karst present someorientation preferences of EastWest and NWSE.Directions to the nearest sinkholes in Devonian Karstpresent a major orientation close to EastWest. Thisorientation preference is consistent in the two con-centrated zones (with a minimum 700 m DNN) in

Devonian Karst (Fig. 12).Nearest neighbor index calculated from DNN indi-cates that the majority of the sinkhole distributionschange from clustered to random or regular patternsalong with the zone scale changing from 700 m to400 and 200 m. This trend is also revealed in the direc-tions to the nearest sinkholes. Figure 13 illustrates rosediagram plots of directions to the nearest sinkholes innested concentrated zones in GalenaMaquoketa karst.Each smaller concentrated zone is within a larger zone.

Fig. 8 Lognormal P P plot for distances to the rst to the thirdnearest sinkholes in Galena-Maquoketa Karst

b

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When the zone scale becomes smaller, the major orien-tation preference becomes more evident.

Results and discussion

Clark and Evanss ( 1954 ) test of sinkholes in the threeactive karst areas indicates that sinkholes tend to beclustered in each active karst area. Maps of sinkholedensity and sinkhole distribution in southeastern Min-nesota display some highly concentrated zones of sink-holes, especially in the active karst areas of Devonianand GalenaMaquoketa Karsts. These areas of con-centrated sinkholes were delineated based on the meanand standard deviation that corresponds to the DNNslognormal distribution.

Sinkholes are highly concentrated in GalenaMa-quoketa and Devonian Karst. In GalenaMaquoketaKarst, approximately 30% sinkholes concentrate in only2% of the total active karst area, the 200-m zones inFig. 10. About 90% sinkholes concentrate in 10% of thetotal karst area, the 400-m zones in Fig. 10. In Devo-nian Karst, more than 70% sinkholes concentrate inonly 3% of the total active karst area, the 700-m zonesin Fig. 10.

The scale effect on sinkhole distribution has beeninvestigated in other karst areas in the U.S. Zhou et al.(2003 ) conducted orientation analysis of sinkholes alongI-70 highway near Fredrick, Maryland. The analysisdemonstrated that orientations of sinkhole pairs at dif-ferent scales correspond to regional and local structuressuch as the axis of regional syncline, the strike of rockunits, major joints, and secondary set of joints.

Analyses of sinkhole distribution in different zoneswithin an active karst area, and directions to the nearestneighbor indicate that sinkhole distributions changefrom clustered to random to regular when the minimumDNN of the concentrated zone decreases. Figure 14illustrates that directions to the nearest sinkholes inmany highly concentrated zones (200 m zone) haveevident orientation preference, especially for zoneswithin which sample size is less than 200. One hypothesisthat can explain this pattern in southeastern Minnesota

Fig. 9 Lognormal P P plot for distances to the rst to the thirdnearest sinkholes in Prairie du Chien Karst

b

Table 7 Mean and standard deviation of distances to the nearestsinkholes derived from lognormal distribution

Karst group MeanrA (m)

rA + r rA + 2 r rA + 3 r

Devonian 67 168 417 1,036GalenaMaquoketa 69 126 233 429Prairie du Chien 134 655 3,215 155,771

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Table 8 Distances to the nearest sinkholes in concentrated zones in Devonian, and Galena-Maquoketa Karst areas

Devonian Karst:

Minimum DNN (m) Zone# Area (km 2 ) N d (no./km 2 ) R c Pattern

700 226 9.23 188 20.37 0.59 ) 10.85 Cluster400 1,288 2.05 110 53.66 0.74 ) 5.23 Cluster700 119 12.2 119 9.75 0.82 ) 3.75 ClusterGalena-Maquoketa Karst:200 1,217 10.31 706 68.48 1.18 9.38 Regular

200 1,273 1.74 130 74.71 1.08 1.78 Random200 1,409 2.86 174 60.84 1.25 6.29 Regular200 1,576 2.18 221 101.38 1.13 3.80 Regular200 1,968 7.55 461 61.06 1.19 7.82 Regular200 1,947 1.64 125 76.22 1.16 3.45 Regular200 1,935 5.31 355 66.85 1.14 4.97 Regular400 640 8.71 295 33.87 0.90 ) 3.13 Cluster400 850 3.77 111 29.44 1.00 ) 0.09 Random400 868 16.7 791 47.37 1.02 1.30 Random400 886 22 724 32.91 0.92 ) 4.20 Cluster400 891 2.72 163 59.93 0.66 ) 8.37 Cluster400 959 4.44 145 32.66 0.88 ) 2.69 Cluster400 996 5 128 25.60 1.01 0.19 Random400 998 7.56 347 45.90 0.97 ) 1.07 Random400 1,036 3.62 140 38.67 0.82 ) 4.15 Cluster400 1,108 8.56 186 21.73 0.90 ) 2.51 Cluster

400 1,138 8.03 250 31.13 0.89 ) 3.24 Cluster400 1,183 28.4 948 33.38 0.95 ) 3.08 Cluster400 1,189 11.8 467 39.58 0.93 ) 2.88 Cluster400 1,193 2.59 126 48.65 0.95 ) 1.08 Random400 1,258 7.51 220 29.29 0.89 ) 3.19 Cluster700 85 99.9 2175 21.77 0.86 ) 12.74 Cluster700 104 4.5 172 38.22 0.52 ) 12.10 Cluster700 42 12.2 296 24.26 0.78 ) 7.08 Cluster700 134 13.2 203 15.38 0.77 ) 6.35 Cluster700 158 19.9 256 12.86 0.71 ) 8.84 Cluster700 175 85.8 1935 22.55 0.85 ) 13.04 Cluster

Fig. 10 Concentrated zones of sinkholes in southeasternMinnesota

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is that structural patterns in these zones favor sinkholeformation. Structural patterns such as the densities andorientations of fractures, joints, and conduit systemsshould be systematically investigated in these areas.Lineament analysis of sinkhole distribution and com-parison between structural pattern and sinkhole distri-bution need to be conducted in the near future to reach aconclusive explanation of sinkhole distribution insoutheastern Minnesota.

Nearest neighbor analysis and sinkhole density dis-tribution in southeastern Minnesota illustrate that somehighly concentrated sinkholes are in the middle of lessconcentrated sinkholes. In many concentrated zones,sinkhole density decreases gradually from the center to

Table 9 Distances to the nearest sinkholes in the nested zones in GlenaMaquoketa Karst

Minimum DNN (m) Zone# Area (km 2 ) N d (no./km 2 ) R c Pattern

700 104 4.5 172 38.22 0.52 ) 12.10 Cluster400 891 2.72 163 59.93 0.66 ) 8.37 Cluster700 42 12.2 296 24.26 0.78 ) 7.08 Cluster400 640 8.71 295 33.87 0.90 ) 3.13 Cluster700 134 13.2 203 15.38 0.77 ) 6.35 Cluster400 996 5 128 25.60 1.01 0.19 Random700 158 19.9 256 12.86 0.71 ) 8.84 Cluster400 1,108 8.56 186 21.73 0.90 ) 2.51 Cluster700 85 99.9 2175 21.77 0.86 ) 12.74 Cluster400 850 3.77 111 29.44 1.00 ) 0.09 Random400 868 16.7 734 43.95 1.08 4.26 Regular200 1,217 10.31 665 64.50 1.27 13.17 Regular400 886 22 640 29.09 0.97 ) 1.24 Random200 1,273 1.74 108 62.07 1.21 4.20 Random200 1,409 2.86 162 56.64 1.29 7.15 Regular400 998 7.56 236 31.22 1.04 1.18 Random200 1,576 2.18 169 77.52 1.20 5.00 Regular400 959 4.44 111 25.00 0.94 ) 1.23 Random400 1,036 3.62 94 25.97 0.91 ) 1.76 Random700 175 85.8 1935 22.55 0.85 ) 13.04 Cluster400 1,183 28.4 876 30.85 0.99 ) 0.30 Random200 1,968 7.55 430 56.95 1.25 9.84 Regular400 1,189 11.8 429 36.36 0.98 ) 0.76 Random200 1,935 5.31 327 61.58 1.19 6.74 Regular400 1,193 2.59 110 42.47 1.06 1.16 Random200 1,947 1.64 109 66.46 1.30 5.92 Regular400 1,138 8.03 240 29.89 0.93 ) 2.20 Cluster400 1,258 7.51 195 25.97 0.94 ) 1.54 Random

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

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

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

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