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In: Sinkholes and the Engineering and Environmental Impacts of Karst (L.B. Yuhr, E.C. Alexander, Jr., and B.F. Beck, eds.), Proceedings of the Eleventh Multidisciplinary Conference, Geotechnical Special Publication no. 183, American Society of Civil Engineers, p.243-256. Predicting Sinkhole Susceptibility in Frederick Valley, Maryland, Using

Geographically Weighted Regression

Katarina Z. Doctor1, Daniel H. Doctor2, Barry Kronenfeld1, David W.S. Wong3, and David K. Brezinski4

1Department of Geography, George Mason University, 4400 University Drive MS 1E2, 22030, Fairfax, VA,

katarina@doctor.com, bkronenf@gmu.edu 2U.S. Geological Survey, 12201 Sunrise Valley Drive, MS 926A, Reston, VA 20192, dhdoctor@usgs.gov 3Earth Systems and GeoInformation Sciences, George Mason University, 4400 University Drive, Room 330

Research I, 22030, Fairfax, VA, dwong2@gmu.edu 4Maryland Geological Survey, 2300 St. Paul Street, Baltimore, MD 21218, dbrezinski@dnr.state.md.us ABSTRACT

A dataset of 556 collapsed sinkholes covering six 1:24,000 scale geologic quadrangles

was analyzed in order to map the relative likelihood of sinkhole formation in Frederick Valley, Maryland, USA. Factors that help predict the density of sinkholes included clustering of sinkholes, geologic structure, rock type, and proximity to: quarries, water bodies, streams, roads, faults, axes of synclines or anticlines, and depth to groundwater. Spatial statistical analyses (K-function, Geographically Weighted Regression (GWR) and Inverse Distance Interpolation) were performed to calculate sinkhole density potential within the study area. K-function analysis was performed to find the clusters over different spatial scales. Using these results sinkhole density was determined for each sinkhole location. Relations of external factors to the calculated interdependent sinkhole density were then examined using Geographically Weighted Regression. The result is a map of sinkhole susceptibility that considers geologic, hydrologic, and anthropogenic factors. The results show that the proximity to the groundwater table, proximity to fold axes, proximity to faults, and proximity to quarries are the factors that significantly influence new sinkhole development, in order of decreasing significance. These results may be a guide for future development activities in this region and others like it. INTRODUCTION

Urban development is rapidly occurring in the karstic terrain of the Frederick Valley, Maryland, USA. Sinkholes are an inherent feature of the landscape; they present hazards and engineering challenges to the development of residential, commercial and industrial infrastructure. A dataset of 556 sinkholes that had evidence they have been collapsed and other geologic features covering six 1:24,000 scale quadrangles was analyzed for this study (Fig. 1).

        

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Figure 1. The study area is Frederick Valley, Maryland. Geologic quadrangles are 1) Catoctin Furnace, 2) Woodsboro, 3) Frederick, 4) Walkersville, 5) Point of Rocks and 6) Buckeystown (from Brezinski, 2004, p. 5, Fig. 1).

Based upon the work of Brezinski (2004), karstic geologic units within the study area were identified and merged in the study region shown in Figure 2. Of the karst features within the study region, 556 collapse sinkholes were identified (Brezinski, 2004).

Figure 2. The karstic region in Frederick Valley, Maryland with the sinkhole distribution.

1  2

3 

5 

4

6 

        

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PREVIOUS STUDIES

Brezinski (2004) prepared a detailed geologic map and karst features description of Frederick Valley, Maryland, and used this information to point out some factors that contribute to sinkhole formation and to identify the geologic units most susceptible to sinkhole development. He noted that human activities such as changing surface stream drainage, storm-water management areas, active quarrying, and soil overburden removal, additionally induce sinkhole development.

Zhou et al. (2003) noted that the location of sinkholes along the I-70 highway in Frederick, MA is irregular, although clustered. The clustering indicates the existence of an interaction between the sinkholes. They concluded in their study that new sinkholes will most likely develop within a 30 m radius of an existing sinkhole.

Orndorff et al. (2000) tested if the distribution of dolines (depressions) is random or not using a statistical method defined by Davis (1986). They also calculated the number of dolines in each geologic unit and indicated the geologic units with higher percentages of dolines. Using focal sum analysis they were able to classify the clustering of dolines into six categories by relating the occurrence of doline clusters to geologic information such as joint orientations and depth to the water table. Finally, they compared surface slope with the location of dolines and found that there are more dolines on flat areas than on slopes.

Magdalene & Alexander (1995) analyzed the distribution of sinkholes in Winona County, Minnesota using nearest neighbor analysis and concluded that the sinkhole distribution is clustered. They also noted that new sinkholes do not develop clustered around old, existing sinkholes, but around other newly developed sinkholes. However, on the scale of several kilometers new sinkholes develop around existing, old sinkholes.

In their study of northern Shenandoah Valley karst Hyland et al. (2006) used weighted overlay analysis in ArcGIS for identifying sinkhole susceptibility. They analyzed the relationship of sinkhole development with four factors: bedrock type, soil depth to bedrock, proximity to geologic faults, and proximity to surface streams. METHODOLOGY

The goal of this study is to assess the influence of both geologic and anthropogenic factors on the formation of sinkholes in the Frederick Valley. The analytical approach involved four statistical methods. First, K-function cluster analysis was performed to depict clusters of point distribution over different spatial scales. Second, the sinkhole density was calculated around each sinkhole location using the results of the K-function analysis to define the buffer size around each sinkhole location. Third, Geographically Weighted Regression analysis was used to identify the factors that significantly influence sinkhole development and to calculate the weights for each sinkhole location taking into account all the influencing factors. Finally, Inverse Distance Interpolation was used to produce a surface map of sinkhole density potential. The general methodology we used is outlined in the flow chart in Fig. 3.

        

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Figure 3. Flow chart of the basic methodology used in this study. CLUSTERING

Cluster analysis is important for analyzing sinkhole development in order to objectively identify patterns and spatial associations with other geographic features (Rogerson, 2001). Nearest neighbor analysis (NNA) is the most common method of cluster analysis; however, this method will not capture easily a pattern that is non-homogeneous over a study area without using higher order statistics which is tedious to calculate (Wong and Lee, 2005). A statistical method that is easier able to identify clustering at different scales is K-function analysis, which compares the observed distribution of sinkholes to the same number of randomly distributed points (Fig. 4). K-function

The K function enables us to depict a cluster of point distribution over different spatial scales. The analysis includes “all points and the distances between all points, not just the nearest neighbors”, and as such “can pinpoint where the clustering or dispersion is most acute over the geographic scales” (Wong and Lee, 2005).

The K-function is defined as (1)

(1)

(Data acquisition and preparation.)

(Define karstic area in the study region.)

(Identify the main bandwidths or scales of clustering.)

(Calculate the sinkhole density.)

(Calculate the shortest distances to the factors that influence sinkhole collapse.)

(Perform Geographically Weighted Regression.)

(Creating a predicted density map.)

Data

Define the study area

Clustering

Density

Proximity to influencing factors

Smooth surface

Regression analysis

        

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where A is total area, N is total number of points, wi is weight at ith point, dij is the distance between i and j points and Ih is indicator for I = 0 when dij > h and I = 1 when dij < h. The “h” is the spatial lag which is the unit reflecting the change in the spatial scale for each sinkhole point. The K(h) function is essentially the point density at a given spatial lag h (Wong and Lee, 2005). Comparing with the point density function of a random point pattern gives the difference function L(h) stated as

(2)

Higher L(h) values indicate a clustering tendency (Wong and Lee, 2005). DENSITY

Sinkhole density is an important factor for determining the area most prone to sinkhole development. Where a closed depression has collapsed into a sinkhole the underlying subsurface contains unstable voids, and possibly a cave. In areas where active sinkholes have developed there is a greater possibility that a new sinkhole will form (Brezinski, 2004; Zhou et al., 2003).

White (1988) estimated sinkhole density by dividing the total number of sinkholes in the karst area with the area of karst. He separately expressed the sinkhole area ratio by dividing the sum of the catchment areas of individual sinkholes with the total area of karst. Brezinski (2004) combined these two equations and calculated the sinkhole potential index (SI) where karstic features include recently collapsed sinkholes (active sinkholes), dolines with no evident sinking throat and springs.

(3)

With both calculation methods the density is expressed for a particular geologic unit; however, the shape and size of each geologic unit can vary greatly thus making direct comparisons between units difficult. Also, in this density calculation one should assume that the density in a particular unit is homogenous, which is unlikely to be a valid assumption.

The method we used to determine the sinkhole density takes into account a radial buffer area around each sinkhole and is expressed as the number of sinkholes inside each buffer zone for which bandwidths (lag spacing) were determined with the previously described K-function analysis. As a result each sinkhole receives an attribute that is equal to the density of sinkholes within its surrounding radial buffer zone. This result is used later for further calculations for estimating sinkhole susceptibility through a map of predicted density.

        

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GEOGRAPHICALLY WEIGHTED REGRESSION

Geographically Weighted Regression (GWR) is a robust technique for examining how the relations between dependent and independent variables change within an area of interest. GWR is a multivariate regression which assumes that the associations of weights of each influencing factor vary across space. Normally in regression, the relation between the dependent and independent variable (or variables, in the case of multivariate regression) is assumed to be constant in space; that is, the relation is assumed to be equally applicable throughout the area of interest (Fotheringham et al., 2002). For a variety of reasons, this is often not the case. For example, one may perform a spatial regression on a number of factors that may relate to the formation of a particular sinkhole, such as rock type, soil type, depth to bedrock, proximity to bedrock structures, local hydrologic factors, resulting in a calibrated model. However, the parameters of the regression model for that particular sinkhole are unlikely to be the same for every other sinkhole in the karst region because the spatial arrangement of the factors influencing sinkhole formation is not uniform (Panno et. al., 1993; McCoy et.al., 2007). GWR allows a regression model to be calibrated separately for every sinkhole, and for the parameters of the resulting numerous regression models to be explored within the area of interest. GWR results are also sensitive to variations in the scale of analysis. Points further away will have less impact on the dependent variables than points closer.

Several geologic and anthropogenic factors were identified in Frederick Valley that influence sinkhole development and collapse. These factors include proximity of sinkholes to quarries, water-bodies, fault lines, anticline and syncline fold axes, streams, roads and ground-water table depth (Zhou et al., 2003; Brezinki, 2004; Magdalene and Alexander, 1995; Orndorff et al., 2000; Hack, 1965). The locations of these features are known from detailed geologic maps (Brezinski, 2004). Geographically Weighted Regression (GWR) analysis was used to express the significance of these factors in relation to sinkhole density. Comparing with the equation of the global regression,

yi = β0 + Σk βk xik + εi , (4) GWR estimates local parameters by calculating the weight for each location:

yi = β0 (ui, vi) + Σk βk (ui, vi) xik + εi , (5) where (ui, vi) are the coordinates of the ith point in the observed space (Fotheringham et al., 2002). Therefore a sinkhole-influencing factor located closer to an observed sinkhole will get higher weight than one further away.

For performing GWR the GWR 3.0 software package was used and the results were imported into ArcInfo 9.2.

INVERSE DISTANCE WEIGHTS INTERPOLATION

The most common method for generating a smooth surface is kriging, which is a

statistical probabilistic method. In this study, a surface map was produced from the GWR

        

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results for predicting the highest sinkhole density using the Inverse Distance Weights spatial interpolation technique (Fig. 5).The Inverse Distance Weights spatial interpolation method is a deterministic process for assigning values to unknown points by using values of a scattered set of points assuming that near locations will have more influence on the predicted value than farther location, weighted inversely to the distance. The predicted value y in location s0 is defined as

(6)

where

(7)

and

(8)

is inverse distance in point i between si and the predicted location s0. RESULTS Nearest neighbor analysis

The nearest neighbor analysis showed that the sinkhole distribution in the Frederick Valley region is highly clustered. K-function analysis

Higher L(h) values indicate the clustering tendency distances from each individual sinkhole (Wong and Lee, 2005). The graph in Figure 4 shows the L-function of sinkhole points calculated based on equations (1) and (2). The peaks reflect the scales at which clustering of sinkhole points is most likely. The clustering is strongest at distances of 1000 to 8000 meters. This suggests the scale at which factors controlling sinkhole development operate. This answer was used to define the buffer for density calculation.

        

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Figure 4. Scatter plot for lag (L) of K–function where the peaks show the cluster sizes. Sinkhole density

Based on resulting values for clustering tendency the value of the highest peak (1600 meters) was chosen to be the buffer size for calculating the sinkhole density. The resulting value was used as a base for the further regression analysis. Geographically Weighted Regression analysis

The F-test of GWR results indicates that the seven analyzed anthropogenic and geologic factors (independent variables) account for a significant portion of the total variation in sinkhole density.

The results of the GWR analysis shows that the t value is < |tcritical | for distance to streams and distance to roads only. Therefore all the other factors are significantly influence the sinkhole development. These factors are: the proximity to quarries, water-bodies, fault lines, fold axes and to groundwater table.

        

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Figure 5. The map of the t-value for proximity to syncline and anticline fold axis.

Figure 6. The map of the parameters for proximity to syncline and anticline fold axis.

        

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Figure 7. The map of predicted sinkhole density.

CONCLUSION

The approach taken in this study was to incorporate many factors that influence sinkhole development into a single, spatially explicit model. The determination of sinkholes’ spatial distribution and pattern (clustering and density) was assessed and the results used to regress against external factors that influence sinkhole development (geology, hydrology and anthropogenic factors) within a GWR model. The result shows that sinkhole formation is not random, and can generally be predicted by external factors. The factors that we found to significantly influence sinkhole collapse development are the proximity to the groundwater table, to syncline and anticline fold axes, to faults, to quarries, and to water-bodies, in decreasing order of significance.

We did not find a statistically significant influence of proximity to roads or to streams on sinkhole development. However, the abundance of these features might make any such significance difficult to detect without separately analyzing the different levels of roads. Higher level roads, such as highways, might influence sinkhole development differently than unpaved or other lower lever roads.

Using the results of the GWR model, a map of the surface prediction of sinkhole density was produced (Fig. 7). Although this map does not predict the exact locations of future sinkholes, it does show which areas are more or less prone to sinkhole development. This information might be useful for planning purposes, including the design of building and road construction standards and the allocation of resources to sinkhole repair activities.

This method is applicable to other areas where sinkhole collapse occurs; however, one must consider that the occurrence of sinkhole collapse is highly sensitive to the local karst environment. Therefore, the factors used in the model may be different than those used in this study. Individual expertise and judgment are necessary to apply the method successfully.

        

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REFERENCES Brezinski, D.K. (2004). “Stratigraphy of the Frederick Valley and its Relationship to

Karst.” Rep. No. 75., Annapolis: Maryland Department of Natural Resources. Fotheringham, A.S., Brunsdon, C., and Charlton, M., 2002. “Geographically Weighted

Regression: the analysis of spatially varying relationships”. Chichester, West Sussex: John Wiley and Sons, Ltd. 269 pp.

Hack, J.T. (1965). “Gomorphology of the Shenandoah Valley, Virginia and West Virginia, and origin of the residual ore deposits.” U.S. Geological Survey Professional Paper 484, 84 pp.

McCoy, K.J. and Kozar, M.D. (2007). “Use of Sinkhole and Specific Capacity Distributios to Assess Vertical Gradients in a Karst Aquifer.” Environ Geol, DOI 10.1007/s00254-007-0889-1.

Magdalene, S. and Alexander, E.C., Jr. (1995). “Sinkhole Distribution in Winona County, Minnesota Revised.” Proceedings of the Fifth Multidisciplinary Conference on Sinkholes and the Engineering and Environmental Impacts of Karst. Gatlinburg, Tennessee.

Orndorff, R.C., Weary, D.J. and Lagueux, K.M. (2000). “Geographic Information Systems analysis of geologic controls on the distribution of dolines in the Ozarks of South-Central Missouri, USA.” Acta Carsologica, Vol. 29- No. 2, 11.

Panno, S.V., Wiebel, C.P., Heigold, P.C. and Reed, P.C. (1993). “Formation of Regolith-collapse Sinkholes in Southern Illinois: Interpretation and Identification of Associated Buried Cavities.” Environmental Geology No. 23, pg. 214-220.

Rogerson, P.A. (2001). “Statistical Methods for Geography.” London, Sage Publications, 232 p.

White, W.B., 1988. “Geomorphology and Hydrology of Karst Terrains”. New York: Oxford University Press, 464 pp.

Wong, D.W.S., and Lee, J. (2005). “Statistical Analysis of Geographic Information with ArcView GIS and ArcGIS.” Hoboken, New Jersey, John Wiley & Sons, Inc.

Zhou, W., Beck, B.F., and Adams, A.L. (2003) “Application of Matrix Analysis in Delineating Sinkhole Risk Areas along Highway (I-70 near Fredrick, Maryland).” Environmental Geology, No. 44, pg. 834-842.