Water table depth, the distance from land surface to the top of the surficial aquifer, is an important map layer, parameter, and input variable for a wide variety of environmental models and decision support methodologies: maps of the water table are used to estimate ground water flow direction and velocity; regional water table maps are important components of ground water vulnerability assessments; ground water depth is an important variable in many hydrologic and pollution transport models; and regional maps of ground water depth are commonly used in environmental decision making such as locating landfills and wastewater disposal sites.
Because water table depth is measured at point locations (in wells or soil borings), water table mapping requires hydrogeologically appropriate techniques to generalize the point measurements. The creation of water table maps is a well accepted practice in ground water investigations - water levels are measured in a network of wells and a surface is interpolated between measuring points. However, this practice works only if the investigated area is small or numerous measuring points are available. Methods are not well established for mapping water table depth over large areas where measurements are sparse and opportunistic.
As part of a ground water vulnerability mapping project for the State of North Carolina, three approaches were considered for regional water table mapping. The first approach uses deterministic modeling, the second uses statistical modeling, and the third uses landscape classification. The three approaches are described in this paper; implementation of the landscape classification method chosen for mapping is discussed in the conference paper, "A Cognitively-based Approach for Hydrogeomorphic Land Classification using Digital Terrain Models" (Fels and Matson 1996).
Ground water, by definition, is hidden from view beneath the land surface. Monitoring wells provide the only direct means of observing ground water. Virtually all we know about ground water is a product of applying and testing hypotheses to generalize observations and measurements conducted in the field or in the laboratory. Rules, models, or schemas for spatially and temporally generalizing monitoring (sample) data across the ground water system are inherent and essential to hydrogeologic science. Our understanding of ground water is the product of a long history of hypothesis and model development, testing, and refinement.
Models employed in hydrogeology span a range of sophistication, abstraction, and rigor, from conceptual models of general hydrogeologic relationships, to deterministic models developed from partial differential equations representing conservation of mass and energy in the ground water system. Any spatial representation of ground water properties or phenomena requires models of some sort to spatially and temporally generalize points measurements.
The water table is the surface of saturated conditions, below which all pores and voids in soil, sediment, or rock are completely filled with water (Heath 1989). Depth to the water table is measured in wells screened across the zone of water table fluctuation (Figure 1). The position of the water table is the product of a wide range of static and dynamic environmental conditions and processes affecting the rate at which water enters and leaves the saturated zone. The water table rises if the rate of water added (recharge) exceeds the rate of water leaving (discharge); conversely, the water table falls if discharge exceed recharge. The water table surface is therefore not static, nor flat (as the name implies), but responsive to climatic, vegetative, geomorphic, and geologic conditions.
Figure 1. Schematic cross-section of a typical surficial aquifer system in humid regions. Solid arrows represent saturated flow, dashed arrows represent unsaturated flow (After Freeze and Cherry 1979).
Traditional water table mapping uses graphical methods to interpolate between water table measurements and hydrogeologic boundaries, with professional judgment and experience filling the gaps in sampling. Computer assisted approaches may incorporate surface mapping methods such as trend surface interpolation and kriging; many of these tools are currently provided in GIS software. Other methods employ mathematical modeling to predict water table elevation from hydrogeologic conditions and processes.
Surface interpolation and mathematical modeling require a high degree of site-specific knowledge and observations. Although detailed water table information may be available in site-specific investigations, sparse and opportunistic water table information must frequently be used in regional hydrogeologic assessments. Interpolation methods do not work well in these situations because they define only general (large wavelength) changes in water table elevation, missing the local (short wavelength) variations in water table elevation with respect to land surface. New developments in spatial analysis and the availability of digital environmental data for large areas provide a number of possible solutions to overcome these regional mapping problems.
As part of a statewide assessment of ground water vulnerability using the DRASTIC method (Aller et al. 1987), the Groundwater Section in the North Carolina Department of Environment, Health, and Natural Resources, required a state-wide map of average annual depth to the water table. The goal of this effort was to use available information to develop a predictive map of average annual water table depth, at a scale of 1:250,000, in a short time-frame, and with a limited budget. Project staff considered using a GIS combined with either deterministic models, statistical models, or landscape classification models in order to accomplish this task. The issues involved in implementing these various models in a GIS are discussed below. The implementation of the landscape classification approach chosen for state-wide water table mapping is discussed separately in the conference paper "A Cognitively-based Approach for Hydrogeomorphic Land Classification using Digital Terrain Models" (Fels and Matson 1996). References in this paper to GIS and spatial analysis assume raster, or grid-cell systems.
Nearly 100 years ago King (1899, as cited in Domenico and Schwartz 1993) observed that the water table is a subdued replica of the land surface in humid regions (Figure 1). This conceptual model is among the oldest in hydrogeology. In North Carolina, water table depth varies widely from just beneath land surface in much of the Coastal Plain to over 100 feet beneath high peaks in the Blue Ridge Mountains. The water table is expressed at land surface where it discharges into rivers, streams, canals, lakes, wetlands, and other surface water bodies throughout the state. At a coarse level, Heath (1988) divides ground water occurrence in North Carolina into two regions of distinct hydrogeologic conditions: the Coastal Plain, and the Piedmont and Blue Ridge Mountains (Figure 2).
Figure 2. The major physiographic regions of North Carolina. Each region has different hydrogeologic characteristics.
The Coastal Plain hydrogeologic region is characterized by sequences of sedimentary deposits overlying metamorphic and igneous bedrock (Heath 1988). The landscape is relatively flat and low in elevation, shallowly dissected by a network of rivers and streams. The Coastal Plain may be further divided on the basis of geomorphology and geologic materials into the Sandhills, the Upper Coastal Plain, the Lower Coastal Plain, and the Coastal Islands (Figure 2). In the Lower Coastal Plain and Coastal Islands, the water table is at or very close to the land surface; wetlands and hydric soils are particularly abundant. The water table is deeper (approximately 5 to 15 feet) beneath modern and relict sand dunes, and on sandy scarps and ridges. Upland areas in the Sandhills and the western Upper Coastal Plain have the deepest water levels although it is unusual for depths to exceed 20 feet. Water levels are shallowest close to streams, in floodplains, and on wide, flat interstream divides (Daniels et al. 1984).
The Piedmont and Blue Ridge Mountain regions are hilly to mountainous and underlain by saprolite and alluvial sediments over bedrock (Heath 1988). Ground water occurs in both the sediments and underlying fractured bedrock. Wetlands are less common than in the Coastal Plain but can occur near streams and in floodplains where the water table is close to or at the land surface. The depth to the water table deepens at the edge of floodplains, where the slope of the land surface increases and alluvial materials give way to colluvial sediments and saprolite. Saprolite, sediment derived from the in-situ weathering of underlying bedrock, is generally thickest at the tops of hills and ridges in the Piedmont and the water table is deepest in these locations at approximately 40 feet (Daniel 1989). In the Blue Ridge Mountains, alluvial sediments are generally coarser, slopes are generally steeper, and saprolite is thin or absent on the slopes, ridges, and mountains. Water table depths are shallow near streams but deepen with increasing slope and topographic relief. Water table depths on inhabited hills and ridges where wells are installed average 60 feet (Daniel 1989) but depths beneath high narrow ridges and peaks exceed 100 feet.
Water table mapping requires a combination of hydrogeologic and ancillary geographic information. Quality controlled, hydrogeologic information available to assist in North Carolina mapping include ambient ground water monitoring data, drilling logs, stream gauge data, and hydrogeologic reports. Digital geographic data for North Carolina include: 1:250,000 scale DEMs, 1:100,000 scale streams and surface water bodies, 1:250,000 scale ground water discharge areas, 1:1,000,000 scale ground water recharge rates, 1:250,000 scale Physiographic Provinces, 1:250,000 scale geology, and a 1:500,000 scale hydrogeologic map of the Piedmont and Blue Ridge Mountains.
Deterministic approaches to ground water modeling employ mathematical equations to explicitly represent the physical relationships, behavior, and properties of the ground water system. Deterministic models use the ground water flow equation, a second-order partial differential equation incorporating Darcy's law of fluid flow and the equation for conservation of fluid mass for three-dimensional flow through porous media (Freeze and Cherry 1979). Governing equations are derived from this equation by making assumptions about the hydrogeologic properties of the system under investigation. By defining boundary conditions, these equations are simplified and solved analytically or numerically. Model results are sensitive to the assumptions, parameters, and boundaries by which the system is defined. The accuracy of mathematical modeling methods are therefore inherently constrained by knowledge of the hydrogeologic parameters and the reasonableness of the simplifying assumptions.
Deterministic water table mapping in a GIS requires linking an analytical or numerical model to calculate a water table elevation surface. This surface must be subtracted from land surface elevation to create a map of depth to the water table.
Analytical ground water models employ closed-form mathematical solutions to the ground water flow equation (Freeze and Cherry 1979). One such solution is the water profile equation (Equation 1, Jacob 1943) which may be used to determine hydraulic head, h in an aquifer between two surface water bodies:
(1)Where h is the height of the water table above the bounding surface water bodies, w is recharge to the aquifer, T is the aquifer transmissivity (hydraulic conductivity times aquifer thickness), a is the distance from a surface water body to the ridge line (or one-half the total distance between water bodies), and x is the horizontal distance to the nearest surface water body.
Key assumptions made to arrive at this solution are that: groundwater flows only in a horizontal direction perpendicular to the surface water bodies, the aquifer is infinite in extent parallel to the surface water bodies and homogeneous and isotropic in its material properties, and the surface water bodies are parallel and at constant and equal elevation (the Dupuit-Forchheimer assumptions for unconfined, saturated ground water flow (Freeze and Cherry 1979)). These boundary conditions and assumptions are most closely approached on North Carolina's coastal barrier islands. C. E. Jacob (1943) first developed and employed this model to simulate the water table profile on Long Island, New York.
Implementation of this model in GIS for average annual water table mapping requires surface water hydrography to determine boundary conditions and digital elevation data to determine surface water elevation, the datum from which water table elevation (h) is calculated. To remain consistent with model assumptions, average annual recharge and aquifer conductivity are constant throughout the mapped region. The height of the water table is determined for a particular grid cell by calculating its distance from the nearest surface water body, the total distance between surface water bodies at that point, and then applying the water profile equation through grid algebra or database calculations. The model is calibrated by comparing field observations with model results and adjusting hydraulic conductivity or recharge rates within acceptable limits. In the absence of either recharge or conductivity data, inverse modeling may be employed to determine these parameters from field measurements of the water table.
Numerical models employ algebraic approximations to solve ground water flow equations. Finite difference numerical methods discretize the continuous ground water flow field into discrete blocks, or elements, within which hydrogeologic properties are assumed constant. The cell-based discretization used in finite-difference methods allows coupling of numerical models with raster GIS software.
Ground water flow in a homogeneous, isotropic aquifer with hydraulic conductivity K, under conditions of uniform recharge w, and assuming horizontal flow is represented by Equation 2 (Domenico and Schwartz 1990):
(2)where h is water table height and x and y are the horizontal dimensions. Letting u = h^2, the numerical approximation for Equation 2 is (Domenico and Schwartz 1990):
(3)where i,j denotes a particular cell in row i and column j in a regular grid with cells of equal x and y dimensions, and delta x is the grid cell width. Boundary conditions must be specified as h^2. Equation 3 calculates hydraulic head as the average of the four adjacent cells plus the constant term relating recharge and conductivity.
Equation 3 can be implemented in a raster GIS as a customized digital filter. Boundary conditions, which must remain fixed (i.e., unfiltered), are set as the squared elevations of surface water bodies. The remainder of the grid is given an intermediate value and Equation 3 is iteratively applied until succeeding values of calculated head converge within a predetermined limit (Domenico and Schwartz 1990). Model results are calibrated from field observations of water level and the conductivity and recharge parameters are adjusted within acceptable ranges.
Deterministic models provide the advantage of explicit solutions to water table elevation, but they are not straightforward to implement in a GIS. Deterministic model applications require adherence to the assumptions made to arrive at solutions to the ground water flow equation. Although the specification of boundary conditions (surface hydrography) is relatively straightforward, model assumptions would most likely be violated for regional mapping because of the spatial variability of hydrogeologic parameters in the surficial aquifer. Deterministic models are usually implemented on site-specific scales where a high level of control is possible for assumptions, parameterization, and calibration. Numerical models are probably best implemented using customized digital filtering techniques in image processing software. The requirement for tuning the model through calibration is essential, but difficult to implement in GIS over large areas with many data points. Incorporating the spatial variation in aquifer properties through tightly coupled GIS and deterministic model linkages requires such substantial overhead in algorithm development that most researchers find loosely coupled linkages to external deterministic models more appropriate (Stuart and Stocks 1994). Nonetheless, considerable potential exists for developing better methods of coupling deterministic models with GIS (Benjamin Houston 1995, personal communication). David Maidment (1996, personal communication) reports success in implementing equation 3 in a GIS as a linked ground water/surface water model. For the state-wide water table mapping exercise at hand, the development time required to establish model linkages and prepare the data sets was prohibitive.
Although the water table is a subdued replica of land surface, the relationship between the two surfaces is not simple, but subject to terrain, geology, climatic conditions, soil properties, and human influences. These relationships are difficult to incorporate and quantify deterministically on a regional scale. A statistical modeling approach to water table mapping seeks to determine which environmental variables are the best predictors of water table depth without explicitly characterizing the physical processes involved, and establishes a mathematical function for the relationships. In statistical models, the explanatory variables may be causative or simply useful estimators of the response variable (Clarke 1994); readily measured environmental characteristics may thus serve as surrogates for hydrogeologic variables in a statistical model. Similar techniques have been employed by Fels (1994) to predict vegetative diversity, and by Gessler et al. (1995) to predict soil properties, from terrain variables extracted from DEMs.
Developing statistical models for water table mapping begins with conceptualizing relationships between water table depth, the response variable, and other mappable characteristics, the explanatory variables. In the North Carolina Piedmont and Mountains, water table depth generally increases with increasing distance from streams as well as with increasing relative and absolute elevation Daniel (1989). In the Coastal Plain, water table depth increases with increasing relative elevation and distance from streams and then it shallows to the land surface on wide interstream divides (Daniels et al. 1984). One or more terrain measurements, or geomorphometrics, may thus serve as explanatory variables for water table depth.
Following the notation of Clarke (1994) the general form of a statistical model for water table depth may be written as:
(4)where WT is water table depth; s is the spatial domain of the function; Elev, Slope, Dist, etc. are the explanatory measurements or values we can extract from continuous data sets in the model domain, such as elevation, slope, and distance to stream; theta is the set of fixed parameters which are estimated from the data; and epsilon is the random component representing fluctuations about the systematic component, f (...). The goal of model development is to determine the most appropriate form of the function f (...), the best estimates for the parameters theta, and the estimated probability distribution of epsilon (Clarke 1994).
The simplest form of the statistical model is the case in which Equation 4 is univariate and linear in the parameters. For example, the simplest form of a statistical model for water table depth could be one in which depth is explained by elevation (Elev) within an acceptable distribution of epsilon:
(5)Equation 5 is a linear regression equation in which the parameters alpha and beta are constants. Inspection of Equation 5 reveals that water table depth (WT) will be zero for only one value of elevation (negative alpha/beta). This simple hypothetical model can only hold in a restricted spatial domain where surface water bodies are all at the same elevation. Wider model application and performance would likely be achieved with the incorporation of additional explanatory variables and the use of more complex model forms such as quadratic, logarithmic, and gaussian.
To develop a statistical model for water table depth and implement it in a GIS, the largest possible set of explanatory spatial variables should be assembled for locations with water table measurements. The explanatory variables initially considered may include any number of geomorphometrics, including elevation, slope, distance to stream, and meso- and micro-scale landform. Data visualization is useful in determining possible explanatory variables. The set of water table measuring locations must be fully representative of the spatial domain: an assumption is made as to which variable explains most of the water table variation and sample locations are randomly distributed within strata of this variable. Statistical methods are employed to determine the form of the equation and estimate the parameters for each of the explanatory variables and the random component. This results in a model equation which directly predicts water table depth.
The model is implemented in a GIS by calculating the relevant explanatory variables for all cells in the model domain. The model equation is applied using grid algebra or database functions to calculate water table depth for each grid cell from the explanatory variables in the same location.
The statistical approach to water table mapping can take into account more of the spatial variability in average water table depth on a regional scale than is generally possible with deterministic models. Statistical models are also rather simple to couple and implement in a raster GIS. On the other hand, existing water level data points may be inadequate or insufficiently stratified for model development, requiring additional monitoring locations to be established, an important restriction if only opportunistic data is available. Other than experimental applications in Mississippi by the US Geological Survey (Chuck O'Hara 1994, personal communication), the statistical approach to water table mapping has not been widely applied. The greatest limitation of statistical models, and the strength of deterministic models, is in simulating the response of the water table to short-term environmental stresses. Statistical modeling is useful for mapping long-term averages but it is generally not adequate for modeling events such as drought, excessive precipitation, or pumping in a well field (David Evans 1996, personal communication).
The previous methods generate a continuous surface of average annual water table elevation or depth. This is the most widely useful spatial representation of the water table because it allows determination of the ground water gradient, an important parameter or variable in many models and hydrologic assessments. However, another mapping approach is the classification of the landscape into ranges of average annual water table depth.
Although land and water table surfaces are highly complex continua, they may be classified into regions of less variability. In the same manner that pedologists use directly observable geomorphic, climatic, geologic, and vegetative characteristics to interpret and map soil properties (Daniels and Hammer 1992), hydrogeologists use similar environmental clues to predict water table depth in the field. This approach seeks to map the water table indirectly by classifying discrete landforms with predictable ranges of average annual water table depth. The resulting classification model is an algorithmic formalization of the conceptual models used in traditional field mapping.
Classification requires rules for spatial discrimination; automated classification requires codable rules for discriminating cells in gridded data sets. A variety of measurements may be calculated from a DEM to represent the landscape characteristics of a particular cell. These measurements, called geomorphometrics, are local, focal, and zonal functions which describe the geometry and topology of cells in the DEM. Landscape settings, or land types, may be defined by logically combining these metrics. In some cases, land types may be defined with a single metric -- cliffs are regions in which slopes exceed a particular value. In other cases, a combination of metrics is required -- ridges have low slope values, but they occur at high relative elevations, and although they are continuous features, elevation may change considerably along their length. The boundaries of land types are often subjective. For example, at what slope value does a ridge become a side-slope? R. Dikau (1989) seeks to define a comprehensive geomorphic taxonomy based on geomorphometrics for landscape features of various scales and for various purposes. In this approach, we seek to define only those mesoscale landforms (i.e., ridges, slopes, valleys, etc.) which reflect water table depth classes.
The classification model is developed in three steps: first, the thematic classes of water table depth are specified; then the land types which reflect these classes are determined; lastly, the land types are interactively classified from a DEM.
For the current study, a hierarchical, 2-level, land type classification scheme was developed to reflect the depth to water table ranges in the DRASTIC methodology (Aller et al. 1987). Within each of North Carolina's six physiographic regions (Figure 2), land types were selected to reflect the prescribed depth ranges (Fels and Matson 1996). Water table depth ranges chosen for the thematic classes should be selected on the basis of the planning or modeling need at hand (the DRASTIC ranges do not necessarily reflect a relevant discrimination of water table depth). It was found that logical combinations of two metrics, slope and landscape position index (a focal function which measures the relative position of a cell with respect to its surrounding cells (Fels 1994)), allow discrimination of the mesoscale land types in the classification scheme. For example, in the Sandhills all cells with landscape position values less than -.005 were classified as hills and ridges (landscape position index is positive in valleys, neutral in mid-slope position, and negative on ridges); of the remaining cells, those with slope values of 2% or greater were classified as sideslopes and those having slopes less than 2% were classified as flats (Fels and Matson 1996). The more diverse features in the Piedmont and Mountains required a more complex classification procedure using the same metrics. Deciding where to break land type classes requires repeated parameterization and visualization of the results. Landscape position and slope values are varied until the best (most cognitively acceptable) fit is achieved between the classified land types and knowledge of water table depth.
Classified depth to water table ranges offer a valuable means of discriminating land areas for certain planning, assessment, and modeling purposes. Although a thematic map layer is not as widely useful as a continuous surface, landscape classification allows water table mapping without the resources required by the deterministic and statistical approaches. Landscape classification can assist field scientists by extending their knowledge into new areas using visual representations of conceptual models for ground water occurrence. Because this approach does not make explicit use of water table measurements, classification accuracy is subject to the knowledge of water table and geomorphic relationships by those conducting the mapping. Classification accuracy is readily measured by cross-tabulating water table depth measurements with the classified ranges and statistically analyzing the amount of agreement. The results of this analysis may be used to adjust the algorithm by which one or more landtypes are discriminated.
Automated regional water table mapping is a complex exercise. The three approaches discussed here illustrate both the value and limitations of using GIS for the task. The most significant value of GIS is the ability to use extensive environmental data sets to parameterize models and establish boundary conditions over large areas. Although some of the tools required to conduct water table mapping are available in raster GIS or image processing packages, all three approaches require customized software for data analysis, model linkages, and export to statistical packages.
Deterministic models offer explicit definition of a water table surface from physical principles and the ability to simulate the response of the water table to short-term environmental stresses. Deterministic models are difficult, however, to implement reliably at regional scales. Statistical models permit direct prediction of water table depth from readily measured explanatory variables without the need to characterize the physical properties or processes in the ground water system. Statistical models also incorporate explicit measures of accuracy, an important (but often missing) component in mapping and modeling activities. Both deterministic and statistical models are data intensive, and generally require new data collection for model development, implementation, and calibration. Landscape classification is an intuitively appealing and efficient approach which formally characterizes conceptual models for water table depth in terms of geomorphic features. Although classification accuracy is measurable, it is subject to the hydrogeologic and geomorphic knowledge of those developing the model algorithms.
Landscape classification was chosen for initial state-wide water table mapping in North Carolina because of its short development time, low implementation cost, and the ability to incorporate existing hydrogeologic knowledge. Classification of land types by hydrogeologic significance supports future model applications by providing a structure for stratifying monitoring locations.
Regional water table mapping is a promising area of research in the geographic and hydrogeologic sciences. The mapping methods presented here will have greater predictive accuracy as they are implemented, compared, and refined in other locations. Further work will also lead to new methods of quantifying regional ground water properties, enhancing the protection of ground water resources. We hope this discussion encourages others to apply and evaluate these water table mapping approaches in their own work.
The authors wish to thank the Groundwater Section of the North Carolina Department of Environment, Health, and Natural Resources, and the United States Environmental Protection Agency, for the opportunity and funding to conduct this work. We thank Carl Bailey, Ted Mew, Arthur Mouberry, and Perry Nelson for conceptualizing and supporting efforts to characterize the hydrogeology of the shallow aquifer system in North Carolina. We also thank Ralph Heath, consultant, and Bruce Lloyd, Charles Daniel, Tim Spruill, and Gerry McMahon of the United States Geological Survey for their insights into regional water table mapping and the approaches discussed here. We are also indebted to Scott Huffman and Kai-Ping Wang of the Groundwater Section for their technical support in this mapping project. Lastly, we are grateful to David Evans, Casson Stallings, and Benjamin Houston for their review of an earlier manuscript of this paper.
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Kris C. Matson
Research Associate
North Carolina State University
Dept. of Biological and Agricultural Engineering
Box 7625, Raleigh, NC 27695-7625
Ph: 919-515-6792, Fax: 919-515-6772
matson@eos.ncsu.edu
John E. Fels
Research Associate Professor
North Carolina State University
Dept. of Landscape Architecture
Box 7701, Raleigh, NC 27695-7701
Ph: 919-515-7341, Fax: 919-515-7330
fels@unity.ncsu.edu