Chapter 1
Distributed Hydrologic Modeling Using GIS
Excerpt from the book Distributed Hydrologic Modeling Using GIS,
by Baxter E. Vieux, published by Kluwer, Dordrecht, the Netherlands,
forthcoming in 2000/2001.
Considering the spatial character of parameters and precipitation
controlling hydrologic processes, it is not surprising that Geographic
Information Systems (GIS) have become an integral part of
hydrologic studies. The primary motivation of this book is to bring
together the key ingredients necessary to use GIS to model hydrologic
processes, i.e., the spatial and temporal distribution of the inputs and
parameters controlling surface runoff. GIS maps describing
topography, land use/cover, soils, rainfall, and meteorological
variables may become model parameters or inputs in the simulation of
hydrologic processes.
Difficulties in managing and efficiently using spatial information
have prompted hydrologists either to abandon it in favor of lumped
models or to develop more sophisticated technology for managing
spatial data (Desconnets et al., 1996). As soon as we embark on
simulating hydrologic processes using GIS, we must address the
issues that are the subject of this book.
1.1
Why Distributed Parameter Modeling?
Historical practice has been to use lumped representations because
of computational limitations or because sufficient data was not
available to populate a distributed model database. How one
represents the process in the mathematical analogy and implements it
in the hydrologic model determines the degree to which we classify a
1
2
Chapter 1
model as lumped or distributed. Several distinctions on the degree of
lumping can be made in order to better characterize a mathematical
model, the parameters/input, and the model implementation.
Whether a model is lumped or distributed depends on whether the
domain is subdivided. It is clear that this is relative to the domain. If
the watershed domain is to be distributed, the model must subdivide
the watershed into smaller computational elements. This process often
gives rise to lumped sub-basin models that attempt to represent
spatially variable parameters/conditions as a series of sub-basins with
average characteristics. In this manner, almost any lumped model can
be turned into a semi-distributed model. Drawbacks associated with
sub-basin lumping include:
1. The model is not physics-based, and
2. Sub-basin lumping may turn out to be extremely cumbersome to
handle the data for a large number of sub-basins.
Sub-basin lumping was an outgrowth of the concept of hydrologically
homogeneous subareas. This concept centered on overlaying areas of
soil, landuse/cover, and slope attributes producing sub-basins of
homogeneous parameters. Sub-basins then could logically be lumped
at this level. Whether hydrologically homogeneous areas can be
justified depends on the uniform nature of many spatially variable
parameters. For example, the City of Cherokee, Oklahoma suffers
repeated flooding from storms having return intervals on the order of
2-year frequency falling on Cottonwood Creek (Figure 1.1). A
lumped sub-basin approach using HEC-HMS
1. Distributed hydrologic modeling using gis
3
Figure 1.1. Contour map of the City of Cherokee in North-western Oklahoma and
Cottonwood Creek draining through town.
(HEC, 2000) is represented schematically in Figure 1.2. ‘Junction-2’
is located where the creek crosses Highway 64 on the Northwestern
outskirts of the City limits. Each sub-basin must be assigned a set of
parameters controlling the hydrologic response to rainfall input.
Practitioners are just starting to profit from research into the
development of distributed hydrology (ASCE, 1999). As distributed
hydrologic models become more widely used in practice, the need for
scientific principles relating to spatial variability, temporal and spatial
resolution, information content, and calibration will become more
apparent.
Though contour lines are the traditional way of mapping
topography, distributed hydrologic modeling requires a digital
elevation model. The Cottonwood basin represented using a 60-m
resolution digital elevation model is seen in Figure 1.3. A distributed
4
Chapter 1
Figure 1.2. HEC-HMS sub-basin definitions for the 125 km2 Cottonwood Creek.
Figure 1.3. Hillshade digital elevation model and road network of the City of Cherokee
and surrounding region.
1. Distributed hydrologic modeling using gis
5
approach to modeling this watershed would consist of a grid
representation of topography, precipitation, soils, and landuse/cover.
1.2
Distributed Model Representation
Figure 1.4 shows a schematic for classifying a deterministic model of
a river basin. Deterministic is distinguished from stochastic in that a
deterministic river basin model estimates the response to an input
using either a physics-based or a conceptual mathematical
representation. Conceptual representations usually rely on some type
of linear reservoir theory to delay and attenuate the routing of runoff
generated. Runoff generation and routing are not closely linked and
therefore do not interact. Physics-based models use equations
conservation of mass, momentum, and energy to represent both runoff
generation and routing in a linked manner. Following the left-hand
branch in the tree, the distinction between runoff generation and
runoff routing is somewhat artificial, because they are intimately
linked in most distributed model implementations. However, by
making a distinction we can introduce the idea of lumped versus
distributed parameterization for both overland flow and channel flow.
A further distinction is whether overland flow or subsurface flow is
modeled with lumped or distributed parameters. Routing flow through
the channels using lumped or distributed parameters distinguishes
whether uniform or spatially variable parameters are applied in a
given stream segment.
Hybrids between these branches exist. For example, the model
TOPMODEL (Beven and Kirkby, 1979) simulates flow through the
range of hillslope parameters found in a watershed. The spatial
arrangement is not taken into account, only the distribution of values
in order to develop a basin response function. It is only a semidistributed model since the statistics of the spatially variable
parameters are operated on without regard to location. TOPMODEL
falls somewhere between conceptual and distributed with some
physical basis.
Temporal lumping occurs with aggregation over time of such
phenomena as stream flow or rainfall accumulations at 5 minute,
hourly, daily, 10-day, monthly, or annual time series. Hydrologic
models driven by intensities rather than accumulations are more
sensitive to temporal resolution. A small watershed may be sensitive
to rainfall time series at 5 minute intervals, whereas a large river basin
may be sensitive to only hourly or longer time steps.
6
Chapter 1
Figure 1.4. Model classification according to distributed versus
lumped treatment of parameters.
Changing spatial resolution of datasets requires some scheme to
aggregate parameter values at one resolution to another. Resampling
involves taking the value at the center of the larger cell. If the center
of the larger cell happens to fall on low/high value, then a large cell
area will have a low/high value. Resampling rainfall maps can
produce erratic results as the resolution increases in size, as found by
Vieux and Farajalla (1996). For the basin and storms tested, as the
resolution exceeded 3 km, the simulated hydrograph became erratic
because of the resampling effect. Resampling is essentially a lumping
process which, in the limit, a single value for the spatial domain
results. How to determine which resolution is adequate for capturing
the essential information contained in a parameter map for simulating
the hydrologic process is taken up in Chapter 4. Farajalla and Vieux
(1995) and Vieux and Farajalla (1994) applied information entropy to
1. Distributed hydrologic modeling using gis
7
infiltration parameters and hydraulic roughness to discover the
limiting resolution beyond which little more was added in terms of
information. Over-sampling a parameter or input map at finer
resolution may not add any more information either because the map,
or the physical feature, does not contain additional information. Of
course variations exist physically, however, these variations may not
have any impact at the scale of the modeled domain.
Numerical solution of the governing equations in a physics-based
model employs discrete elements. The three representative types are
finite difference, finite element, and stream tubes. At the level of a
computational element, a parameter is regarded as being
representative of an average process. Thus, some average property is
only valid over the computational element used to represent the
process of flow. For example, porosity is a property of the soil
medium, but it has no meaning at the level of the pore space itself.
From a model perspective, a parameter should be representative of
the surface or medium at the scale of the computational element used
to solve the governing mathematical equations. This precept is often
exaggerated as the modeler selects coarser grid cells, losing physical
significance. In other words, runoff depth in a grid cell of 1 km
resolution can only be taken as generalization of the actual runoff
process and may or may not produce physically realistic model
results.
Computational resources are easily exceeded when modeling large
basins at fine resolution, motivating the need coarser model
resolution. One of the great questions facing operational use of
distributed hydrologic models for large river basins is how to
parameterize sub-grid processes. At the scale of more than a few
meters in resolution, runoff depth and velocity have little physical
significance. Depending on the areal extent of a river basin and the
spatial variability inherent in each parameter, small variations may not
be important. Can physically realistic behavior be expected from a
model that uses such coarse resolution as to have lost physical
significance? We will see in Chapter 6 how hydraulic roughness may
be inferred from landuse/cover at the watershed scale. Chapter 4 deals
with resolution issues related to information content. Chapter 10 takes
up the issue of adjusting parameters for distributed model calibration.
8
1.3
Chapter 1
Mathematical Analogy
Physics-based models solve governing equations derived from
conservation of mass, momentum and energy. Unlike empirically
based models, differential equations are used to describe the flow of
water over the land surface or through porous media, or energy
balance in the exchange of water vapor through evapotranspiration.
Simplifications are made, because the differential equations contain
terms for which the accompanying parameters, boundary or initial
conditions are unknown, or because the resulting nonlinear equations
are difficult to solve. The resulting mathematical analogies are
simplifications of the complete form. The full dynamic equations
describing the flow of water over the landsurface or in a channel
contain gradients that may be negligible under certain conditions. In a
mathematical analogy we discard the terms in the equations that are
orders of magnitudes less than the others. Simplifications of the full
dynamic governing equations give rise to zero inertial, kinematic, and
diffusive wave analogies.
If the physical character of the hydrologic process is not supported
by a particular analogy, then errors result in the physical
representation. Difficulties also arise from the simplifications because
the terms discarded may have afforded a complete solution while their
absence causes mathematical discontinuities. This is particularly true
in the kinematic wave analogy, in which changes in parameter values
can cause discontinuities, sometimes referred to as shock in the
equation solution. Special treatment is required to achieve solution to
the kinematic wave analogy of runoff over a spatially variable surface.
Vieux et al. (1990) and Vieux (1991) presented such a solution using
nodal values of parameters in a finite element solution. This method
effectively treats changes in parameter values by interpolating their
values across finite elements. The advantage of this approach is that
the kinematic wave analogy can be applied to a spatially variable
surface without numerical difficulty introduced by the shocks that
would otherwise propagate through the system. Vieux and Gaur
(1994) presented a distributed watershed model based on this nodal
solution using finte elements to represent the drainage network.
Chapter 9 presents a detailed description of the solution
methodology used by r.water.fea. The naming convention stems from
the original concept of a GIS tool resident within the GRASS GIS for
simulation of surface runoff in watershed. This model employs a
1. Distributed hydrologic modeling using gis
9
kinematic wave analogy solved with finite elements in space and finite
difference in time. This analogy is most suited to watersheds in which
backwater is not important. Such watersheds are usually in the upper
reaches of major river basins where landsurface gradients dominate
flow velocities.
The diffusive wave analogy is necessary where backwater effects
are important. This is usually in flatter watersheds or low-gradient
river systems. Mathematically, the diffusion term smoothes out
numerical discontinuities due to changes in parameters typical in most
watersheds. CASC2D (Julien and Saghafian, 1991; and Julien, et al.,
1995) uses the diffusive wave analogy to simulate flow in a grid cell
(raster) representation of a watershed. This model solves the diffusive
wave analogy using a finite difference grid corresponding to the grid
cell representation of the watershed. The diffusive wave analogy
requires additional boundary conditions to obtain a numerical solution
in the form of supplying a gradient term at boundaries or other
locations. CASC2D essentially uses a more complex mathematical
analogy to overcome numerical difficulties, even though in many
cases watershed conditions do not have flat slopes requiring this
analogy.
1.4
Runoff Processes
Two basic flow types can be recognized: overland flow,
conceptualized as thin sheet flow before the runoff concentrates in
recognized channels, and channel flow, conceptualized as occurring in
recognized channels with hydraulic characteristics governing flow
depth and velocity. Overland flow is the result of rainfall rates
exceeding the infiltration rate of the soil. Surface runoff may be
generated either as infiltration excess or saturation excess depending
on soil type and topography.
1.4.1
Infiltration Excess (Hortonian)
Infiltration excess first identified by Horton is typical in areas
where the soils have low infiltration rates and/or the soil is bare. Rain
drops striking bare soil surfaces break up soil aggregates, allowing
fine particles to clog surface pores. A soil crust of low infiltration rate
results particularly where vegetative cover has been removed due to
farming or fire. Infiltration excess is generally conceptualized as flow
10
Chapter 1
over the surface in thin sheets. Model representation of overland flow
uses this concept of uniform depth over a computational element
though it differs from reality where small rivulets and drainage swales
convey runoff to the major stream channels. Figure 1.5 shows two
zones, one where rainfall, R, exceeds infiltration I (R>I); the other
where R < I. In the former, runoff occurs; in the latter, rainfall is
infiltrated, and infiltration excess runoff does not occur. However, the
amount of infiltrated rainfall may contribute to the watertable,
subsurface conditions permitting. Figure 1.5 is a simplified
representation, since more than two zones are likely present in a
natural watershed. From hill slope to stream channel there may be
areas of infiltration excess which runs on to areas where the
combination of rainfall and run-on from upslope does not exceed the
infiltration rate, resulting in losses to the subsurface.
Simulation of infiltration excess requires soil properties and initial
soil moisture conditions. Figure 1.6 shows two plots: rainfall intensity
as impulses, and infiltration rate as smoothly decreasing with time.
The infiltration rate is a potential rate governed by soil properties and
the initial degree of saturation. Infiltration excess occurs when the
rainfall rate exceeds the infiltration rate. Richards equation fully
describes this process using principles of conservation of mass and
momentum. The Green and Ampt equation (Green and Ampt, 1911) is
a simplification assuming piston flow (no diffusion). Modeling
infiltration excess at the watershed scale requires estimation of
infiltration parameters from mapped soil properties. Loague (1988)
found that the spatial arrangement of soil hydraulic properties at
hillslope scales (< 100 m) were more important than rainfall
variations. Order of magnitude variation in hydraulic conductivity at
length scales on the order of 10 m controlled the runoff response. This
would seem to say that infiltration modeling at the river basin scale is
impossible unless very detailed spatial patterns of soil properties are
known. The other possible conclusion is that not all of this variability
is important over large areas. Considering that detailed measurement
is not economically feasible over large spatial extent, deriving
infiltration rates from soil maps is an attractive alternative. Estimating
infiltration parameters from soil maps and associated databases of
properties is considered in Chapter 5.
1. Distributed hydrologic modeling using gis
Figure 1.5. Schematic diagram of runoff produced by infiltration excess
Figure 1.6. Infiltration excess modeled using the Horton equation.
11
12
Chapter 1
1.4.2
Saturation Excess (Dunne)
Saturation excess runoff is common in mountainous terrain or
watersheds with highly porous surfaces. Under these conditions,
overland flow may not be observed. Runoff occurs by infiltrating to a
shallow watertable. As the gradient of the watertable steepens, more
runoff to stream channels occurs. As the watertable surface intersects
the ground surface areas adjacent to the stream channel, the surface
saturates. As the saturation zone grows in areal extent, and rain falls
on this area, more runoff occurs. Figure 1.7 shows the location of
saturation excess first identified by Dunne.
Figure 1.7. Schematic diagram of runoff produced by saturation
excess.
1.5
GIS Data Structures and Sources
New sources of geographic data, often in easily available global
datasets, offer tantalizing detail if only they could be used in a
hydrologic model designed to take advantage of the tremendous
information content. Unfortunately, hydrologic models have not kept
pace with many new data sources. Once a particular spatial data
1. Distributed hydrologic modeling using gis
13
source is considered for use in a hydrologic model, data structure, file
format, quantization (precision), and error propagation become
important. GIS offers efficient algorithms for dealing with most of this
data. However, the relevance of the data to hydrologic modeling is
often not known without special studies to test whether a new data
source provides advantages meriting its use. Chapter 2 deals with the
major data types necessary for distributed hydrologic modeling.
Depending on the particular watershed, many types of data may
require processing before they can be used in a hydrologic model.
1.6
Surface Generation
Models often require a surface representation of a parameter that is
measured at points. Much work has been done in the area of spatial
statistics and the development of Kriging techniques to generate
surfaces from point data. In fact, several methods for generating a
two-dimensional surface from point data may be enumerated:
 Linear interpolation
 Local regression
 Distance weighting
 Moving average
 Splines
 Kriging
The problem with all of these methods when applied to
geophysical fields such as rainfall, ground water flow, wind, or soil
properties is that the interpolation algorithm may violate some
physical aspect. Gradients may be introduced that are a function of the
sparseness of the data and/or the interpolation algorithm. Values may
be interpolated across distinct zones where natural discontinuities
exist.
Suppose, for example, that several piezometric levels are
measured over an area and that we wish to generate a surface
representative of the pressure within the aquifer. Using an inverse
distance-weighting scheme, we interpolate pressures in a raster array.
We will almost certainly generate a surface that has artifacts of
interpolation that violates physical characteristics, viz., gradients are
introduced that would indicate flow in directions contrary to the
known gradients or flow directions in the aquifer. In fact, a literal
interpretation of the interpolated surface may indicate that, at each
measured point, pressure decreases in a radial direction away from the
14
Chapter 1
well location. None of the above methods ensure physical correctness
in the interpolated surface.
Depending on the sampling interval, spatial variability, physical
characteristics of the measure, and the interpolation method, the
contrariness of the surface to physical or constitutive laws may not be
apparent until model results reveal intrinsic errors introduced by the
surface generation algorithm. Chapter 3 deals with surface
interpolation and hydrologic consequences of interpolation methods.
1.7
Spatial Resolution and Information Content
How resolution in space affects hydrologic modeling is of primary
importance. The resolution that is necessary to capture the spatial
variability is often not addressed in favor of simply using the finest
resolution possible. It makes little sense, however, to waste computer
resources when a coarser resolution would suffice. We wish to know
the resolution that adequately samples the spatial variation in terms of
the effects on the hydrologic model and at the scale of interest. This
resolution may be coarser than that dictated by visual esthetics of the
surface at fine resolution.
The question of which resolution suffices for hydrologic purposes
is answered in part by testing the quantity of information contained in
a data set as a function of resolution. We can stop resampling at finer
resolution once the information content ceases to increase.
Information entropy, originally developed by communication
engineers, can test which resolution is adequate in capturing the
spatial variability of the data (Vieux, 1993). We can relate the
information content to model performance effects. For example,
resampling rainfall at coarser resolution and inputting this into a
distributed hydrologic model can produce erratic hydrologic model
response (Vieux and Farajalla, 1996). Chapter 4 provides an overview
of information theory with an application showing how information
entropy is descriptive of spatial variability lost by resampling to
coarse resolution.
1.8
Drainage Networks and Resolution
Drainage networks are derived from digital elevation models
(DEMs) by connecting each cell to its neighbor in the direction of
principal slope. DEM resolution has a direct influence on the total
1. Distributed hydrologic modeling using gis
15
drainage length and slope. Too coarse resolution causes an
undersampling of the hillslopes and valleys where hilltops are cut off
and valleys filled. Two principal effects result:
1. Drainage length is shortened by short-circuiting,
2. Slope is flattened.
Though these effects on hydrograph response may be compensating,
shorter drainage length accelerates arrival times at the outlet, whereas
flatter slopes delay. The influence of DEM grid cell resolution is
discussed in Chapter 7, and the effect on hydrograph response is
demonstrated in Chapter 11.
Input to the distributed hydrologic model consists of rain and/or
snow. Depending on the climatic zone considered, rainfall may be the
only significant source in the hydrologic cycle. For more mountainous
watersheds, we will need to consider both snow and rain not only
from the standpoint of input but also in order to model runoff from
frozen ground and snowmelt. In this book we focus on rainfall and
runoff resulting from infiltration excess.
1.9
Spatially Variable Precipitation
Besides satellite, one of the most important sources of spatially
distributed rainfall is radar. Spatial and temporal distribution of
rainfall is the driving force for both infiltration and saturation excess.
In the former case, comparing rainfall intensity with soil infiltration
rates determines the rate and location of runoff. One of the most
hydrologically significant radar systems in the US is the WSR-88D
(popularly known as NEXRAD) radar. Understanding how this
system produces rainfall estimates is paramount for deriving accurate
input to hydrologic models. Resolution in space and time, errors,
quantizing (precision), and availability in real-time or post-analysis
will be taken up in Chapter 8.
1.10
r.water.fea: A Distributed Hydrologic Model
The model r.water.fea was developed for the U.S. Army Corps of
Engineers, Construction Engineering Research Laboratory,
Champaign, Illinois (USA-CERL). This model is a part of the
Geographic Information Systems (GIS), GRASS (Geographic
Resource Analysis Support System). A description of the interface
between GIS and the finite element and finite difference algorithms to
16
Chapter 1
solve the kinematic wave equations are examined in detail in Chapter
9. Assembly of finite elements representing the drainage network
produces a system of equations solved in time. The resulting solution
is the hydrograph at selected stream nodes, cumulative infiltration,
and runoff depth in each grid cell.
1.11
Distributed Model Calibration
Once the assembly of input and parameter maps for distributed
hydrologic model is completed, the model must still be calibrated. The
argument that physics-based models do not need calibration
presupposes perfect knowledge of the parameter values and location
as well as rainfall input. This is clearly not the case. Besides the
parameter and input uncertainty, there are resolution dependencies as
presented by Vieux et al. (1993). Lumped modelers have long argued
that there are too many degrees of freedom in distributed modeling
vis-a-vis the number of observations. This concern does not take into
account that if we know the spatial pattern of a parameter, we can
adjust its magnitude while preserving its relative variation in space.
This calibration procedure can be performed manually by applying
scalar multipliers or additive constants to parameter maps until the
desired match between simulated and observed is obtained.
Automatic calibration of non-physics-based models must rely on
optimization such as the shuffled complex evolution method (Duan, et
al., 1992). Physics-based models have the advantage that there are
governing differential equations. This fact may be exploited using the
adjoint technique, which has enjoyed success in meteorology in
retrieving initial conditions for atmospheric models. Chapter 10
covers the manual and automatic calibration of the r.water.fea model.
The fact that we can invert the differential equations in the presence of
data dismisses the concern that there are too many degrees of freedom
and unique solutions are not possible. There is the limitation that the
problem is ill-conditioned, meaning that optimal parameters may exist
but search algorithms may not efficiently retrieve them. In any case, it
is clear that multiple minima do not exist, at least in the cases
examined, in spite of the difficulty of retrieving the optimal solution
automatically.
1. Distributed hydrologic modeling using gis
1.12
17
Case Studies and Concluding Remarks
Chapter 11 contains case studies showing the influence of lumping
spatially variable parameters. If there is no effect due to lumping, the
spatial variation has little or no impact on the simulation results.
Resolution effects on hydrograph response are also demonstrated,
showing influence of drainage length and slope. These case studies are
presented for a series of several storms using radar input for the 2400
km2 Illinois River basin straddling the border between Oklahoma and
Arkansas.
Ideally, this book raises more questions than it answers. Depending
on the reader’s interest, the techniques described should have wider
application than just the subset of hydrologic processes that are
addressed in the following chapter. They have been used to develop
scientific principles of distributed hydrologic modeling using GIS. In
an effort to make the book general, techniques described may be
applied using many different GIS packages. With this framework in
mind, the principles are general to distributed hydrologic modeling
without the restriction to any particular proprietary GIS routine or
software.
1.13
1.
2.
3.
4.
5.
6.
7.
8.
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