gao spatial operation

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O R I G I N A L P A P E R

Spatial operations in a GIS-based karst feature database

Yongli Gao

Received: 31 December 2005 / Accepted: 2 April 2007

Springer-Verlag 2007

Abstract This paper presents the spatial implementation

of the karst feature database (KFD) of Minnesota in a GISenvironment. ESRIs ArcInfo and ArcView GIS packages

were used to analyze and manipulate the spatial operations

of the KFD of Minnesota. Spatial operations were classi-

fied into three data manipulation categories: single layer

operation, multiple layer operation, and other spatial

transformation in the KFD. Most of the spatial operations

discussed in this paper can be conducted using ArcInfo,

ArcView, and ArcGIS. A set of strategies and rules were

proposed and used to build the spatial operational module

in the KFD to make the spatial operations more efficient

and topographically correct.

Keywords Karst feature database (KFD) Single layer

operation Multiple layer operation Map projection

Grid-based transformation

Introduction

Manipulation of spatial data is an essential function in both

GIS (Demers 1997) and spatial information systems (Lau-

rini and Thompson 1992). Chou (1997) divided spatial

operations in GIS into single layer and multiple layer

operations. Barnett (1994) constructed a complex 8 8

spatial transformation matrix in a conceptual digital carto-

graphic generalization model. The development, manage-

ment, and data analyses of the Minnesota karst feature

database are described in a series of papers (Gao et al. 2006;

Gao and Alexander 2003; Gao et al. 2005a, b, c). This paperpresents the spatial implementation of the karst feature

database (KFD) of Minnesota in a GIS environment. Spatial

operations were classified into three data manipulation

categories: single layer operation, multiple layer operation,

and other spatial transformation in the KFD.

Single layer operations, also known as horizontal oper-

ations, apply to only one data layer and provide the most

fundamental tools of data preparation for spatial analysis

(Chou 1997). Single layer operations in the spatial opera-

tion module are divided into three categories: feature

manipulation, feature selection, and feature classification.

Feature manipulation changes the spatial features of a data

layer. Feature selection identifies features by using spatial

manipulation or logical expressions. Feature classification

classifies features into groups.

Multiple layer operations, also known as vertical oper-

ations, concurrently operate on more than one data layer

(Chou 1997). Chou (1997) classified multiple layer oper-

ations into overlay, proximity, and spatial correlation

analyses. In the KFD, proximity and spatial correlation

analyses were built into the spatial analysis module. The

multiple layer operations in the KFD include overlay,

feature selection, and feature classification. Overlay oper-

ations manipulate different spatial data layers to generate

combined spatial features according to logical connections

between data layers (Chou 1997). Feature selection and

feature classification operations are similar to those oper-

ations described in the single layer operations except that

they operate on multiple layers.

Other spatial transformations in the spatial operation

module include digitization and map generalization, pro-

jection, and grid-based transformation. Digitizing and map

generalization include operations about how to generate

Y. Gao (&)

Department of Physics, Astronomy, and Geology,

East Tennessee State University,

Johnson City, TN 37614, USA

e-mail: [email protected]

123

Environ Geol

DOI 10.1007/s00254-007-0896-2


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georeferenced coverages through on-screen and off-screen

digitization, tabulated data, and scanned images. Trans-

forming three-dimensional space onto a two-dimensional

map is called projection (ESRI 1994). Map projection in

the spatial operation module involves defining map pro-

jections for data layers and conversions between different

types of map projections. In GIS and spatial information

systems, spatial data are commonly represented by twodifferent data models, vector and raster data models.

Vector data model represents the points, lines, and areas of

geographical space by exact X and Y coordinates. In the

raster data model, space is represented as a continuous

surface that is divided into grid cells. Most of the spatial

operations discussed above focus on the vector data

structure. Grid-based transformation includes operations

involving one or more grid data layers.

ESRIs ArcInfo and ArcView GIS packages were used

to analyze and manipulate the karst feature database

management system (DBMS). Similar operations in other

GIS packages may be different. Many other GIS packageshave normally identical tools. The same tool in different

GIS packages, even in ArcInfo and ArcView can give

different results.

Single layer operations

Single layer operations apply to only one data layer in GIS.

GIS-based spatial operations implemented three kinds of

single layer operations on the karst feature DBMS and

related geological and geographic data layers. These

operations are feature manipulation, feature selection, and

feature classification.

Feature manipulation

Feature manipulation on a single GIS data layer usually

changes the spatial objects in the data layer. Operations

such as add, delete, move, split, eliminate, dissolve, and

buffer were conducted on single data layers in the KFD.

Add, delete, and move

Any individual spatial object can be added, deleted, or

moved from the GIS data layer. For instance, users can use

the applications of karst feature DBMS built in ArcView

GIS to add, delete, or move point features such as sink-

holes, springs, and stream sinks in a GIS data layer or a

GIS-based database.

Any selected points, lines, and polygons can be removed

from the data layer in an intuitive way in both ArcView and

ArcInfo. For example, in ArcView GIS, if a data layer is in

edit mode, any selected objects can be deleted by pressing

the delete key of the keyboard. Delete command can be

used in Arc/Info to remove certain objects.

Adding new spatial features in a data layer is straight-

forward in both ArcView and ArcInfo. Drawing tools can

be used to draw different spatial objects in a georeferenced

environment. Clicking the pointer tool once creates a point

at the location of the pointer. A line segment, called arc inGIS, is a basic unit for both line and polygon coverage. The

arc-node data model is widely used in modern GIS to

represent and produce arcs and polygons. In this model, an

arc is connected by two nodes, a start node and an end

node, and zero or any number of vertices in between. The

difference between a node and a vertex is that a node has

topological meaning besides geographic coordinates. A

polygon is an area connected and closed by individual line

segments. In ArcInfo and ArcView, drawing a line segment

can be done by double clicking at the starting point (the

start node), single clicking a series of points (vertices), and

double clicking at the last point (the end node). The ver-tices define the shape of a line segment. A polygon is added

by continuously drawing a series of line segments and

ending at the starting point to close the polygon. A node in

a polygon is a start node of one line segment and an end

node of the other. To be topologically correct, a polygon

must be closed.

All spatial objects can be moved easily if the data layer

is in editing mode. Moving the nodes and vertices of a line

segment or polygon can change the shapes and locations of

these objects.

Split and merge

A split operation is conducted by drawing straight lines

through existing line segments or polygons to split them.

Any polygon or line segments through which the straight

line passes would be split into smaller line segments or

polygons.

The merge operation is the opposite of split. It combines

selected adjacent line segments or polygons to form a new

longer line segment or larger areas.

Eliminate

The eliminate operation is commonly used to remove un-

wanted sliver polygons. Sliver polygons are very small

polygons along the boundary of normal polygons. In many

cases, they are invisible at normal scales. Sliver polygons

may occur as a result of map overlay, mapjoin, or building

topography after a map is digitized.

In ArcInfo, the command Eliminate can be used to

merge very small polygons with neighboring polygons that

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have the largest area. With the LINE option, ELIMINATE

merges selected arcs separated by unwanted nodes (called

pseudo nodes in GIS) into single arcs.

The areas of sliver polygons are usually significantly

smaller than normal polygons. They can be selected basedon a threshold value of polygon areas. Any polygons with

areas smaller than that threshold value can be selected and

then removed from the polygon coverage. This procedure

can be achieved in both ArcView and ArcInfo and is very

effective to remove unwanted small polygons. Figure 1

illustrates the removal of sliver polygons along the bounder

between Fillmore and Olmsted Counties after their bedrock

geology layers were combined as one single layer.

Dissolve

The dissolve operation is used to eliminate unwanted

boundaries between adjacent polygons, line segments, and

regions to merge adjacent objects having the same value for

a specified attribute. This operation can be used after

multiple GIS coverages are joined or combined to eliminate

the boundaries between adjacent spatial objects having the

same value for the identifying attribute. The dissolve

operation can also be used to reclassify an existing layer

based on a different classification attribute. Figures 2 and 3

illustrate that the original bedrock geology in Fillmore

County was reclassified into three different bedrock groups:

non-carbonate, Galena-Maquoketa, and Devonian. The

boundaries between adjacent polygons belonging to thesame group were eliminated using the dissolve operation in

ArcView GIS.

Buffer

The buffer operation creates buffer zones around selected

features based on the distances from these features. The

input layer could be any feature type, but the results of

buffer operations are polygon features. Users can specify

the option to keep or to dissolve the boundaries among

intersected buffer zones. The distances of buffer zones

could be a constant distance or a series of equal distance

zones. The equal-distance option can be used to investigate

the relationship of a spatial occurrence and the proximity to

a set of spatial features (Chou 1997). This operation is used

to study sinkhole distribution and to construct sinkhole

probability maps.

Feature identification and selection

Features in a single layer can be identified or selected using

the graphical user interface (GUI) or logical expressions.

Both ArcInfo and ArcView have GUI tools to identify or to

select features in a single data layer.

GUI tools can directly select features from the maps by

moving the pointer to the features to be selected or to

delineate an area to select all features falling in the delin-

eated area. Logical expression such as Structured Query

Fig. 1 Eliminating sliver polygons along the border between

Fillmore and Olmsted Counties after their bedrock geology layers

were combined as one single layer. a Before elimination. b The sliver

polygons were eliminated

Fig. 2 Bedrock geology in

Fillmore County classified by

bedrock formations (data

source: Mossler 1995)

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Language (SQL) is used in ArcView and ArcInfo to select

features based on the values of specified attributes.

Figure 4 shows a query expression used in ArcView GIS to

select all sinkholes from the karst feature index layer.

In addition, users can develop specific GUI tools forfeature identification and selection. Some GUI tools were

developed for the applications built on the karst feature

DBMS to identify and select karst features in ArcView

GIS.

Feature classification

Spatial data are usually classified for spatial analysis or

modeling. A spatial data set can be classified into any

number of classes. The fundamental issue in spatial feature

classification is to determine the number of classes speci-

fied for a spatial theme.

In a single data layer, feature classification can be

specified based on field observation and scientific visuali-

zation, properties and distributions of the specified identi-

fying attributes, and the purpose of the classification.

Figures 2 and 3 show that bedrock geology in Fillmore

County can be classified into individual bedrock forma-

tions or into three major groups with each group including

several bedrock formations. Note that some frequency

distributions have conventional classification formula ormethods. For instance, equal interval or equal frequency

can be used to classify uniformly distributed features or

variables. Mean and standard deviation are used to classify

normally distributed attributes. A bimodal distribution can

be classified into two groups. If the distribution of a feature

shows a very complex pattern, that feature can be classified

into several subclasses, and each subclass can be classified

further based on its distribution within the subclass. This

process can be repeated until the classes and subclasses are

clearly defined. Feature classifications based on different

attributes can also be superimposed to reach a final deci-

sion-making classification. For example, karst feature

classification based on county and bedrock geology can be

superimposed to generate karst groups for each county.

Multiple layer operations

Multiple layer operations manipulate more than one data

layer in a GIS. GIS-based spatial operations implemented

three kinds of multiple layer operations on the karst feature

DBMS and related geological and geographic data layers.

These operations are overlay, feature identification and

selection, and feature classification.

Overlay

Overlay operations on multiple GIS data layers create

combined or mutual-exclusive features from the input

spatial features in different data layers. Operations such as

union, identity, intersect, clip, erase, update, mapjoin, ap-

pend, and edgematch were conducted on multiple data

layers in the KFD.

Fig. 4 Using a logical expression to identify or select features in

ArcView GIS. The example above selects all sinkholes in the karst

index layer

Fig. 3 Karst areas in Fillmore

County generated by feature

classification and the dissolving

operation. Bedrock geology in

Fillmore County was

reclassified into three different

bedrock groups and adjacent

polygons belonging to the same

group were combined using the

dissolve operation

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Union, identity, and intersect

Union, identity, and intersect are the most commonly used

overlay operations in ArcInfo. All three operations super-

impose two GIS layers to compute the geometric inter-

section of the input and overlay coverages. The basic

structure of these three operations can be represented as:

Input theme overlay theme output theme

The main difference among these three operations is the

way in which spatial features from the input coverages are

preserved. The output theme should have the same feature

type as the input theme. For example, if the input theme is

a polygon coverage, the result of the overlay operation will

also be a polygon coverage. The output theme should

contain attributes from both the input and the overlay

themes. The overlay coverage should always be a polygon

theme.

The union operation generates a new theme containingthe features and attributes of two polygon themes. All

polygons from both coverages will be split at their inter-

sections and preserved in the output coverage. The input

theme for the union operation has to be a polygon cover-

age, therefore the output theme will be a polygon coverage

as well.

The intersect operation only preserves the portion of the

input that falls inside the overlay theme. The input theme

can be a point, line, or polygon coverage. The overlay

themes features will split the input theme.

The identity operation preserves all features of the input

theme as well as those features of the overlay theme thatoverlap the input coverage. The input theme can be a point,

line, or polygon coverage. The overlay themes features

will split the input theme.

Users need to select an appropriate operation, based on

what they want to preserve in the output layer. Figure 5

illustrates the use of the union operation in combining

active karst areas from different counties. Intersecting

bedrock geology coverage and areas whose depth to bed-

rock is less than 50 ft (15 m) produced the active karst

areas in each county.

Clip, erase, and update

These three operations extract, erase, or replace a portion

of an input theme using selected or all features from a

polygon theme. The input theme of clip and erase opera-

tions can be a point, line, or polygon coverage. The input

theme of the update operation must be a polygon coverage.

The clip operation extracts a portion of the input theme

using a polygon theme as a cookie cutter. For example,

the Fillmore County outline can be used to clip the karst

feature index coverage to generate all karst features in

Fillmore County.

The erase operation is the opposite of the clip operation.

It erases features from the input theme that overlap with the

erase polygon theme. For example, the Fillmore county

outline can also be used to exclude all karst features from

the karst feature index coverage that fall inside Fillmore

County.

Instead of extracting or erasing a portion of an input

theme, the update operation replaces features from the in-

put theme that overlap with the update polygon theme.Figure 6 illustrates the use of a newly developed bedrock

geology map in replacing the southwestern corner of the

bedrock geology map in the seven-county metropolitan

Twin Cities area, Minnesota.

Mapjoin, append, and edgematch

Mapjoin and append operations combine adjacent cov-

erages into one coverage. The main difference between

mapjoin and append is that the mapjoin operation

recreates topography but applies only to polygon themes.

The append operation applies to point, line or polygon

themes, but the topography needs to be rebuilt after the

operation. Overlay operations discussed above apply to

only two themes, an input theme and an overlay theme.

Mapjoin and append operations can combine up to 500

coverages.

Mapjoin and append operations can generate many er-

rors and discontinuities and are usually followed by

edgematch, dissolve, clean, and build operations to recreate

a clean topography. The edgematch operation is used to

Fig. 5 Generate a combined active karst area using the union and

intersect operations. Active karst areas from different counties were

combined together using the union operation. Active karst areas in

each county were generated by intersecting bedrock geology coverage

(Mossler 1995) and depth to bedrock coverage (Mossler and Hobbs

1995)

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align features along the edges of adjoining coverages.Figure 7 demonstrates that a gap along the Olmsted County

border was aligned and merged using edgematch and dis-

solve operations. The resulting coverage combines 350 m

buffer zones of sinkholes from different counties.

Feature identification and selection

Features from one GIS theme can be identified or se-

lected based on their locations and relationships with

other themes. For example, sinkholes falling outside of

active karst areas can be selected to verify their locations

and the accuracy of the active karst boundaries. Figure 8is a portion of such a selection. The input theme is the

sinkhole coverage and the identifying theme is the active

karst theme in southeastern Minnesota. It first selects

sinkholes falling inside the active karst areas and the

selection is then switched to select sinkholes falling

outside the active karst areas. The selected sinkholes

are then converted to a separate coverage for verification

of sinkhole locations and bedrock geology or depth tobedrock boundaries.

Feature classification

Features of one GIS theme can be classified based on its

spatial relationships with other themes. For example,

sinkholes can be classified based on the underlying bedrock

geology. This can be done by spatially linking the sinkhole

and the bedrock geology coverages and then classifying the

sinkholes based on their bedrock formations.

Other spatial transformations

Some spatial transformations need special procedures or

data models different from those discussed in the single

and multiple layer operations. These operations are digi-

tizing and map generalization, projection, and grid-based

transformation.

Fig. 6 Updating the bedrock geology (Mossler and Tipping 2000) in

the seven-county metropolitan Twin Cities area using the update

operation. a Old bedrock geology map. b A new bedrock geology

map for the southwestern corner of the metropolitan area. c Updated

bedrock geology after the update operation

Fig. 7 The cleanup of

appended coverages using

edgematch and dissolve

operations. a A gap along the

Olmsted County boundary after

combining the 350 m buffer

zones of sinkholes from

counties. b Filling the gap after

edgematching. c Dissolving the

boundary to merge the two

polygons

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Digitizing and map generalization

Digitizing and map generalization include operations onhow to generate georeferenced coverages through on-

screen and off-screen digitization, scanned images, and

tabulated data.

On-screen digitization can be done based on a features

relative location on a background map. Figure 9 shows that

karst outcrops were digitized based on their locations on a

registered topographic map in the Lewiston Quadrangle,

Minnesota in ArcView GIS. The outcrops were originally

mapped and drawn on U.S. Geological Survey (USGS)

1:24,000 topographic maps. Off-screen digitization can be

done by connecting a computer to a digitizer. The map to

be digitized is taped on the digitizer tablet and then reg-istered and digitized from the digitizer. Scanned images

can be registered and rectified in ArcInfo and then traced to

generate a georeferenced coverage.

Digitization and image tracing are techniques to convert

cartographic maps into GIS coverages. Tabulated data can

also be loaded into ArcInfo or ArcView and then converted

into GIS coverages based on their geographic coordinates.

Data stored in the KFD can be linked and converted into

GIS coverages based on the karst features UniversalTransverse Mercator (UTM) coordinates.

Map projection

Transforming three-dimensional space onto a two-dimen-

sional map is called projection (ESRI 1994). Many types

of projections exist and different projections preserve dif-

ferent spatial characteristics such as area, azimuth, and

distance. The different ways of projecting the spherical

earth onto two-dimensional maps result in different dis-

tortions. The UTM and geographic projections are the two

commonly used projections for the karst feature DBMS.During map projection, a spheroid that approximates the

actual earth must be defined. Since North America has

been surveyed many times in the past, many spheroids for

the earth have been defined (ESRI 1994). Spheroids de-

fined by North American Datum 1927 and 1983 (NAD27

and NAD83), and World Geodetic System 1984 (WGS84)

are used on the karst feature DBMS. Map projection in the

karst feature DBMS involves defining map projections for

data layers and conversions between different types of map

projections. The ArcView GIS does not have the capability

to define or convert map projections. The command Pro-

jectdefine and Project in ArcInfo can be used to accomplish

these operations. The following is a projection file asso-

ciated with the Project command to convert a coverage

from UTM NAD27 to UTM NAD83:

input

projection utm

zone 15

datum nad27

parameters

output

projection utm

zone 15datum nad83

parameters

end.

Geographic Calculator, developed by Blue Marble

Geographics (Bell 2000) was used as a coordinate con-

version tool for the KFD. The Arctoolbox in ArcGIS (A

new GIS package from ESRI that combines ArcInfo,

ArcView, and many other tools and extensions) has the

capabilities to define and to convert map projections for a

Fig. 9 Digitize karst outcrops on USGS 1:24,000 topographic maps

in ArcView GIS. The outcrops were originally mapped and drawn on

topographic maps. This example is from the Lewiston Quadrangle,

MinnesotaWinona Co., 7.5 min series topographic map, 1974

Fig. 8 Select sinkholes based on their spatial relationship to active

karst areas. a Sinkholes on top of active karst. b Sinkholes falling

outside of active karst areas were selected and then converted into a

separate coverage

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GIS coverage through a graphic user interface (Tucker

2000). Projecting geographic data from one projection

system to another system usually results in some errors.

Projection errors in the karst feature DBMS generally can

be omitted since the error of the location itself is much

larger than the projection error. In order to maintain a

spatially compatible database, the projections of all the

karst feature data are defined as UTM NAD83. If a set ofGIS data is systematically shifted to different locations, this

may indicate that the projection definition for this set of

data is wrong. This happened to the Winona data set in the

karst feature DBMS. The data set was projected multiple

times and the final projection was defined as UTM NAD83

instead of NAD27. The actual projection was UTM

NAD27. The systematic error was corrected to project the

data set from NAD27 to NAD83.

Grid-based transformation

Grid-based transformations include operations involvingone or more grid data layers. Grid data can be converted

into vector data and vice versa. Spatial relation types of

vector data such as neighborhood statistics, distance to

spatial feature, spatial density, and proximity analysis can

be used to generate spatial analysis and convert the results

of the analysis into a grid. The spatial analyst provides

many GUI tools in ArcView GIS to manipulate and ana-

lyze grid-based data (Ormsby and Alvi 1999). Figure 10 is

Fig. 10 Generating a sinkhole density grid from the sinkhole

coverage in southeastern Minnesota

Bedrock Topography

topogrid

Surface DEM Grid of Bedrock Topography

Depth to bedrock grid

subtract

Shallow bedrock

reselect (< 50 ft.)

Regions of shallow bedrock

region group

Mask layer

set small regions to null

Regions of shallow bedrock

nibble small regions

Nibbled shallow bedrock

Boundary clean

Smoothed shallow bedrock grid

covert to shape file

A polygon coverage of shallow bedrock

Fig. 11 Cartographic model flow chart to generate shallow bedrock

coverage from bedrock topography

Fig. 12 Bedrock topography for the seven-county metropolitan Twin

Cities area (data source: Mossler and Tipping 2000)

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a sinkhole density grid generated from the sinkhole cov-

erage in southeastern Minnesota in ArcView GIS.

Grid-based transformations and analyses are usually

combined with vector-based operations and analyses to

extract useful information and generate desired coverages.

Figure 11 shows a cartographic model flowchart that gen-

erates a shallow (less than 50 ft or 15 m) depth to bedrock

coverage for the seven-county metropolitan Twin Cities

area. The input layer is a line coverage of bedrock topog-

raphy. Figure 12 is the original bedrock topography map

and Fig. 13 is the depth to the bedrock grid. Figure 14

demonstrates the map query and map calculator tools used

to execute processes from reselect to boundary clean as

shown in the cartographic model (Fig. 11) to clean up the

topography of the depth to bedrock grid. Figure 15 is the

final polygon coverage of shallow bedrock.

Discussion

Most of the spatial operations discussed in this paper can

be conducted using both ArcInfo and ArcView. ArcView is

more user-friendly with many GUI tools and extensions.

However, many spatial operations in ArcView GIS result

in some node and intersection errors, many unnecessary

nodes, and incorrect topography. These problems become

more evident if complex data sets are involved in these

operations. Compared with ArcView, ArcInfo is not as userfriendly as ArcView, but usually generates cleaner topog-

raphy and fewer errors. One approach using ArcView and

ArcInfo is to use ArcView to preprocess the data and then

use ArcInfo to correct errors and build topography. Users

can use ArcInfo commands such as nodeerrors, labelerrors,

intersecterr, generalize, clean, and build to correct Arc-

View processed coverages.

Users can also use ArcInfo Macro Language (AML),

ArcView Avenue scripts, and ArcView ModelBuilder to

Fig. 13 Grid of depth to bedrock coverage for the seven-county

metropolitan Twin Cities area

Fig. 14 Map query and map calculator tools used to clean up the topography of the depth to bedrock grid

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automate a series of spatial operations. ArcView applica-

tions written in ArcView Avenue scripts for the karst

feature DBMS automatically conduct many spatial and

analytic procedures. Figure 16 is a working model built in

ArcView using ModelBuilder tools. This model builds

several depths to bedrock grids from 95% randomly se-

lected water wells in Olmsted County using differentinterpolation methods. The model can be saved and mod-

ified for future work. For example, the remaining 5% of the

water wells can be added to the model to evaluate the

accuracy of the different interpolation methods.

Any spatial model used in a GIS should be optimized to

make the spatial operations more efficient and topograph-

ically correct. The following strategies are used to build

spatial operational models in the karst feature DBMS:

1. If a model involves both single layer and multiple

layer operations, single layer operations should be

conducted as early as possible. This can reduce manyerrors and improve the performance of multiple layer

operations.

2. The topography of the result of any spatial operation

should be cleaned and rebuilt. Many spatial operations

result in many errors for the intermediate coverages

and these errors can propagate into future operations.

Some propagation errors are very hard to detect and

may corrupt the GIS data and system if not detected

and fixed earlier.

3. The number of spatial features involved in any spatial

operation should be minimized to exclude unnecessary

features. Feature selection and classification are very

useful approaches to limit the number of spatial fea-

tures for a spatial operation.

4. If attribute tables are also involved in a spatialoperation,

DBMS optimization techniques can be implemented

to maintain data consistency. Combining DBMSs

query optimization and concurrency control techniques

and GISs spatial operational optimization techniques

can result in more efficient and robust spatial data

models.

These strategies and techniques were used to build theDBMS and GIS applications for the karst feature DBMS.

These rules and strategies were also enforced in the spatial

operations for the spatial analyses and probability models

of sinkhole distribution in Minnesota.

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Fig. 15 Shallow bedrock coverage for the seven-county metropolitan

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