Storm Runoff Computation Using Gis

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Storm Runoff Computation Using

Spatially Distributed Terrain Parameters

by Francisco Olivera and David R. Maidment

University of Texas at AustinCenter for Research in Water Resources

1. Introduction

In rainfall-runoff computation, not only is the generation of excess precipitation

spatially distributed but also the precipitation itself, which has been a limitationfor the use of the classic unit hydrograph model for years. The theory presented in

this paper is an attempt to generalize the unit hydrograph method for runoff

response, and to do so on a spatially distributed basis in which the runoff responses from subareas of the watershed are considered separately instead of

being spatially averaged.

Although the theory of linear routing systems presented in this article is not bound

to raster representations of the study area, the model proposed here is based ongrid data structures. A grid data structure is a discrete representation of the terrain

based on identical square cells arranged in rows and columns. Grids are used to

describe spatially distributed terrain parameters (i.e. elevation, land use,impervious cover, etc.), and one grid is necessary per parameter that is to be

represented. The density of grid cells should be large enough to resemble a

continuous character of the terrain.

Starting from the digital elevation model (DEM), hydrologic features of theterrain (i.e. flow direction, flow accumulation, flow length, stream-network, and

drainage areas) can be determined using standard functions included in

commercially available geographic information system software that operates onraster terrain data. At present, DEM’s are available with a resolution of 3 arc-

seconds (approximately 90 m) for the United States, and 30 arc-seconds

(approximately 1 Km) for the entire earth, etc. Since in the case of water draining

under gravity a single downstream cell can be defined for each DEM cell, aunique connection from each cell to the watershed outlet can be determined. This

process produces a cell-network, with the shape of a spanning tree, that represents

the watershed flow system.

Flow routing consists of tracking the water throughout the cell-network. For this

purpose, a two-parameter response function is determined for each cell, in which

the parameters are related to flow time (flow velocity) and to shear effects

(dispersion) in the cell. Flow-path response functions are calculated byconvoluting the responses of the cells located within the reach. Finally, the

watershed response is obtained as the sum of the cell responses to a spatially

distributed precipitation excess.



A deconvolution algorithm is used to estimate the precipitation excess from flow

records instead of from precipitation records. This algorithm consists of

deconvolving an observed hydrograph by an estimated watershed responsefunction (unit hydrograph) to obtain the precipitation excess. The spatial

distribution of the precipitation excess is assumed to be proportional to the runoff

coefficient.

2. Literature Review

Pilgrim (1976) carried out an experimental study consisting in tracing floodrunoff from specific points of a 0.39 Km2 watershed, near Sydney, Australia, and

measuring the travel time of the labeled particles to the outlet. A conclusion of his

study is that "at medium to high flows the travel times and average velocities become almost constant, indicating that linearity is approximated at this range of

flows". Additionally, for a watershed subdivided into non-overlapping subareas,

linearity of the routing system implies that the overall watershed response is equal

to the sum of the responses of its subareas, which is an important insight indealing with spatial variability of the watershed.

An significant attempt to linking the geomorphological characteristics with the

hydrologic response of a watershed is given by Rodriguez-Iturbe and Valdes

(1979). In their paper, Horton’s empirical laws, i.e. law of stream numbers,lengths and areas, are used to describe the geomorphology of the system. Mesa

and Mifflin (1986), Naden (1992) and Troch et al. (1994) present similar

methodologies to account for spatial variability when determining the watershedresponse. The catchment response is calculated as the convolution of a network

response and a hillslope response. The network response is calculated as the

solution of the advection-dispersion equation, weighted according to the widthfunction of the network. However, the researchers present no physically-based

methodology to determine the hillslope response.

An interesting approach to model the fast and slow responses of a catchment is

presented by Littlewood and Jakeman (1992, 1994). In their model, the watershedis idealized as two linear storage systems in parallel, representing the surface and

the subsurface water systems. The surface system is faster and affects mainly the

raising limb of the resulting hydrograph, while the subsurface system is slow anddetermines the tail of the response.

Geographic Information Systems (GIS) are tools that allow one to jump fromlumped to spatially distributed hydrologic models. The border between lumped

and distributed models is not sharp, and there are pre-GIS attempts to deal withspatially distributed terrain attributes. For example, the Hydrologic Engineering

Center (HEC) flood model HEC-1, well known as a lumped model, allows the

user to subdivide the watershed into smaller sub-basins for analysis purposes, androute their corresponding responses to the watershed outlet. In this case, the

concept of purely lumped model does not apply, although it cannot be considered



a fully spatially distributed model either. It is therefore advisable to keep in mind

the extent to which a given model is lumped or distributed.

Grid-based GIS appears to be a very suitable tool for hydrologic modeling,mainly because "raster systems have been used for digital image processing for

decades and a mature understanding and technology has been created for thattask" (Maidment 1992 a). The ESRI Arc/Info-GRID system as well as the U.S.

Army Corps of Engineers GRASS system, work on grid data structures. Gridsystems have proven to be ideal for modeling gravity driven flow, because a

characteristic of this type of flow is that flow-directions depend entirely on

topography and not on any time dependent variable. This characteristic is whatmakes gravity driven flow easy to be modeled in a grid environment and,

consequently, grid systems include hydrologic functions as part of their

capabilities. At present, hydrologic functions, available in GRID and GRASS,allow one to determine flow direction and drainage area at any location, stream

networks, watershed delineation, and others (Maidment 1992 a).

Recently, there have been attempts to take advantage of GIS capabilities for

runoff and non-point source pollution modeling. Vieux (1991) presents a reviewof water quantity and quality modeling using GIS and, as an application example,

employs the kinematic wave method to an overland flow problem. GIS is used to

process the spatially variable terrain and the finite elements method (FEM) tosolve the mathematics. Maidment (1992 a, 1992 b, 1993) presents a grid-based

methodology for determining a spatially distributed unit hydrograph that assumes

a time-invariant flow velocity field. According to him, the velocity time-

invariance is a requirement for the existence of a unit hydrograph with a constanttime base and relative shape. This concept is also explained in this article, in the

light of the conditions for linearity of a routing system. In Maidment's articles,from a constant velocity grid, a flow time grid is obtained and subsequently theisochrone curves and the time-area diagram. The unit hydrograph is obtained as

the incremental areas of the time-area diagram, assuming a pure translation flow

process. A more elaborate flow process, accounting for both translation andstorage effects, is presented by Maidment et al. (1996 a). In their paper, the

watershed response is calculated as the sum of the responses of each individual

grid-cell, which is determined as a combined process of channel flow (translation

process) followed by a linear reservoir routing (spreading process). Olivera et al.(1995) and Olivera and Maidment (1996 a) present a grid-based, unsteady-flow,

linear approach that uses the diffusion wave method to model storm runoff and

constituent transport. In these articles, the routing from a certain location to theoutlet is calculated by convoluting the responses of the grid-cells of the drainage-

path.

Sensitivity of model results to the spatial resolution of the data has been addressed

by Vieux (1993), who discusses how the grid-cell size affects the terrain slope andflow-path length, and, in turn, the surface runoff. Vieux and Needham (1993)



conclude that increasing the cell size shortens the streams length and increases the

sediment yield.

Attempting to account for spatial variability of the terrain in storm runoff modeling, researchers have taken either of the following paths: (1) partitioning the

hydrologic system into subsystems and applying lumped models to each of them,or (2) developing GIS interfaces to generate input files for other lumped models,

and display the results in the form of a map. In both cases, an improvement withrespect to the traditional fully lumped models has been accomplished; however,

these kind of solutions can not be considered spatially distributed. In this research,

storm runoff is modeled within GIS, redefining the use of GIS by using it as amodeling tool itself and not only as a link between the heterogeneous terrain and

an existing non-GIS model.

3. Methodology

For a spatially uniform hydrologic system, the classic unit hydrograph modelstates that

( 1 )

where t [T] is time, Q(t) [L3T-1] is the flow at the watershed outlet, AW [L2] is the

watershed area, I(t) [LT-1] is the excess precipitation, and U(t) [T-1] is the

watershed unit hydrograph. Likewise, for a spatially distributed linear systemsubdivided into uniform non-overlapping subareas, this equation takes the form of

(Maidment et al. 1996)

( 2 )

where NW is the number of subareas, Ai [L2] is the area of subarea i, I i(t) [LT-1] isthe excess precipitation in subarea i (subarea input), and U i(t) [T

-1] is the response

at the watershed outlet yield by a unit instantaneous input in subarea i. Notice that

it is because of the additivity property that characterizes linear systems, that theoverall watershed response can be calculated as the sum of the subarea responses.

From the physical point of view, this summation implies that the flow of a

subarea input to the watershed outlet is not affected by the flow of the other subarea inputs, and that all inputs can be routed simultaneously yet independently.

The use of equation (2) requires for each subarea the excess precipitation Ii(t), and

the response function Ui(t).

In this study, the subareas are taken as small square cells that resemble thecontinuous character of the landscape (see Figure 1), and, because the number of



The methodology consists of: (1) calculating the overall watershed response as

the weighted sum of the cell responses, where the weight is given by the excess

precipitation (see equation (2)); and (2) estimating the spatially distributed excess precipitation by deconvolving the watershed flow records by the watershed

response (see equation (1)), and distributing this lumped excess precipitation

based on the terrain physical characteristics. It becomes clear that both processesare strongly related, and that one cannot be considered without the effect of the

other. Routing the excess precipitation from the terrain to the watershed outlet is

covered in Flow Routing , while estimating the volume and spatial distribution of the excess precipitation in Excess Precipitation.

3.1. Flow Routing

The response at the watershed outlet cell yield by a unit instantaneous input in a

cell is called here flow-path response function Ui(t), and consists of two parts: a flow-path redistribution function U'i(t) [L

-1] that represents the translation and

redistribution processes in the flow-path (lag-time from the cell to the outlet andspreading around the centroid of the mass slug); and a flow-path loss factor K i(t)

[-] that accounts for the losses along the flow-path. Note that, because of howlosses are accounted for, the area under the curve U'i(t) vs. t is equal to one, and

the values of K i(t) are less than one (see Figure 3).



Figure 3: The flow-path response function results from the product of the flow-

path redistribution function and the flow-path loss factor.

Flow-path redistribution functions U'i(t) have to satisfy certain mathematical

properties so that if, for example, an input in cell A is routed to cell B and then to

the cell C, the result should be the same as if it were routed directly from A to C(see Figure 4).

Figure 4: Flow-path from cell A to cell C.



To understand the implications of this condition, notice that the flow-path

redistribution function U'AB(t) is the probability density function of a random

variable XAB that represents the time spent by a water particle in the flow-path thatruns from cell A to cell B. Accordingly, U'BC(t) and U'AC(t) are the probability

density functions of random variables XBC and XAC, respectively. Since the time

spent in AC is the sum of the times spent in AB and BC, it follows that XAC = XAB

+ XBC. In terms of probability density functions, this is expressed as

( 3 )

where * stands for convolution integral. Equation (3) implies that the

redistribution functions should be self reproducing, i.e., the convolution of tworedistribution functions results in a function of the same type. This condition

precludes one, for example, from defining the redistribution functions as

exponential distributions; in other words, from modeling the watershed as an

array of linear reservoirs with one linear reservoir per grid cell, which is a

common but erroneous approach. In statistical terms, the type of functions thatcan be used as redistribution functions are called infinitely divisible distributions.

The normal, gamma and first-passage-times distributions are examples of infinitely divisible distributions.

From the physical point of view, if the flow-path is assumed to convey one-

dimensional unsteady flow and the inertial terms in the St. Venant momentum

equation are neglected, the flow can be modeled with the diffusion wave equation(Miller and Cunge, 1975, Lettenmaier and Wood 1993). Thus, if lateral inflow is

not considered, the flow is represented by

( 4 )

where x [L] is the distance along the flow-path, t [T] is the time, qi(t) [L3T-1] is the

flow at any time and point of the flow-path, C i [LT-1] is the kinematic wave

celerity, and Di [L2T-1] is a dispersion coefficient. For a unit-impulse input, the

solution for qi of equation (4) at the flow-path outlet is the flow-pathredistribution function U'i(t), and it results in a first-passage-times distribution

(Nauman 1981):

( 5 )

where Ti = Li / Ci is the mean value of the distribution (the lag time in the flow-

path), ∆ i = Ci Li / Di represents the spreading around the mean of the distribution

(the shear and storage effects on the flow), and L i [L] is the flow-path length.

First-passage-times distributions apply to systems bounded by a transmitting



barrier upstream (open boundary) and an adsorbing barrier downstream (close

boundary). First-passage-times distributions are in accordance with what other

researchers have proposed to model the time spent by water in hydrologic systems(Mesa and Mifflin 1986, Naden 1992, Troch et al. 1994). Likewise, it has been

shown that other infinitely-divisible two-parameter distributions, say normal or

gamma, do not differ significantly from the first-passage-times distribution withthe same first and second moments (Olivera 1996 b). Olivera (1996 b) has also

observed that three-parameter distributions tend to overestimate the importance of

the tails with respect to the central part of the distribution.

Extending the concept of self-reproducing flow-path redistribution functions tothe cell level, allows one to model the flow based on scale-independent terrain

parameters. Since the time spent in a flow-path is equal to the sum of the time

spent on each its constituting cells, i.e., Xi = x1 + x2 + ... + x N where Xi is arandom variable that represents the time spent in the flow-path and x1, x2, ...and x N

are random variables that represent the time spent in each of the N cells that form

the flow-path, it follows

( 6 )

where U'i(t), u'1(t), u'2(t), ... u' N(t) [T-1] are the probability density functions of Xi,x1, x2, ... and x N respectively. Moreover, because U’i(t) is a first-passage-times

distribution, u'1(t), u'2(t), ... and u' N(t) are also first-passage-time distributions that

can be expressed as

( 7 )

where j refers to the cell of the flow-path, v j [LT-1] is the flow velocity, d j [L2T-1]

is the dispersion coefficient (shear and storage effects), t j [T] is the expected flow

time through the cell and l j = v j t j. Because the cell flow length l j is known, theonly two parameters needed to define u' j(t) are the flow velocity v j and the

dispersion coefficient d j. In some cases, it is preferable to define the

dimensionless Peclet number v jl j/d j - instead of the dispersion coefficient d j - to

describe the shear and storage effects in the cell. However, it should be noted that, because it involves the flow length in its definition, the Peclet number is a scale

dependent parameter.

The connection between the flow-path redistribution function and the cellredistribution functions is given by equation (6). However, the use of this

equation implies as many convolution integrations as cells are there in the

watershed. Depending on the hardware available, this process might be extremely

demanding and time consuming. A good approximation to U'i(t) - whose error falls within the limits of the uncertainty of the model parameters - can be obtained

based on the fact that (DeGroot 1986)



(8)

where E refers to expected value (first moment) and Var to variance (secondmoment around the mean). According to this method, the approximate solution

for U'i(t) has the same first and second moments as the solution obtained with

equation (6). By equalizing the first and second moments of U'i(t) given by

equation (5) to the sum of the moments of the u' j(t) given by equation (7),

relations between Ti and ∆ i, and v j and d j are determined as

( 9 )

and

( 10 )

The main advantage of this approach is that it can be applied automatically by

using standard functions – like the weighted flow length function - included incommercially available geographic information systems software that operates on

raster terrain data.

Water losses in linear systems are represented by a first-order loss term in the

mass balance equation

( 11 )

where Λ i [T-1] is a flow-path loss coefficient. The solution of equation (11) is

given by

( 12 )



where U'i(t) - the flow-path redistribution function - is equivalent to the flow-path

response function when losses are neglected, and exp(-Λ i t) represents the lossesin the flow-path. The flow-path loss factor K i(t) is, therefore, given by

( 13 )

Similarly, for a cell of a flow-path, it can be demonstrated that

( 14 )

where k j(t) [-] is the cell loss factor and λ j [T-1] is a cell loss coefficient.

For small losses, the cell loss factors can be approximated to the constant value k j

= exp(-λ j t j), and the flow-path loss factor to the product of the cell loss factors

( 15 )

which, considering that t j = l j / v j, is equal to

( 16 )

Finally, the flow-path response function can be expressed as

( 17 )

where Ti, ∆ i and K i are given in equations (9), (10) and (16) as functions of the

flow velocity v j, the dispersion coefficient D j and the loss coefficient λ j. Note

that if the cell inter-connectivity and the grids of v j, D j and λ j are defined, the

terrain would be fully described for flow routing purposes.

3.2. Excess Precipitation

One of the advantages of the theory of linear routing systems is that it can handle

spatially distributed inputs, letting the excess precipitation vary according to the

terrain physical characteristics, say land use, soil type or topography.

Standard engineering practice estimates excess precipitation based on soil-water balance. Willmott et al. (1985), for instance, have developed the WATBUG



Fortran program that simulates the soil-water balance based on local temperature,

precipitation and soil water-holding capacity. Although the soil-water balance is a

physically based approach, it has been observed that it is a complicated processthat should account for a large number of parameters, that it is very sensitive to

the data available, and that it produces results that have to be interpreted with

extreme caution. Other simple excess precipitation models such as the SoilConservation Service (SCS) curve number method, or just the product of a runoff

coefficient by the precipitation, are examples of attempts of solving the problem.

However, estimating the correct excess precipitation is still far from beingachieved.

In the following, a deconvolution methodology for determining excess

precipitation from flow instead of from precipitation records is presented. Given

the flow at a specific station, and a spatially-distributed parameter that describesthe terrain tendency to generate runoff, the spatially-distributed excess

precipitation and flow parameters are calculated. The method consists of

deconvolving the observed direct runoff by the watershed unit response function.This unit response is estimated from the flow records, and considers thewatershed as a lumped system. Spatial variability of the terrain is considered later

in the process. The relation between direct runoff and excess precipitation is given

by

( 18 )

where Q(t) [L3T-1] is the direct runoff, A [L2] is the watershed area, r(t) [T-1] is the

estimated watershed unit response, and Pe(t) [LT-1] is the excess precipitation.

Determining the excess precipitation consists of solving equation (18) for Pe. The

excess precipitation is calculated at discrete time steps by trial and error, guessingvalues of Pe for each time step and then verifying if equation (18) is satisfied. An

optimization software helped in the process of determining the excess precipitation values.

Since the flow and unit response function are considered for the watershed as a

whole, the excess precipitation obtained by deconvolution is a lumped type of

result that should be distributed according to the local hydrologic characteristicsof the terrain. It is assumed that the amount of excess precipitation produced by

each cell is a function of the runoff coefficient, a well known hydrologic

parameter that can be estimated from tables available in the literature (Chow et al.

1988, Browne 1990, Pilgrim and Cordery 1993). A connection between excess precipitation in a cell and runoff coefficient is a critical assumption that allows

one to use a simple model without going through a more complicated - but not

necessarily more reliable - soil-water balance. In our model, excess precipitationin a cell Ii is assumed to be proportional to the runoff coefficient ci minus a

uniformly distributed abstractions parameter ζ (i.e., Ii = ci - ζ or 0 whichever is

greater). This abstractions parameter ζ constitutes a threshold value, andaddresses the fact that low-developed areas might yield no runoff at all. The



abstractions parameter ζ may change from event to event depending on rainfall

intensity, soil infiltration capacity and antecedent moisture condition. Values of ci

- ζ for each cell are calculated and used as an excess precipitation scale factor.The excess precipitation generated in a cell Ii is given by

( 19 )

where

( 20 )

A j is the area of cell j, and subscript i refers to the cell where the excess

precipitation is being calculated. Note that the values of α i have an average value

of one.

To calculate the spatially-distributed flow parameters (flow velocity vi and

dispersion coefficient Di), the watershed unit response, used for the deconvolution

in equation (18), is equalized to the weighted sum of the flow-path responses

( 21 )

in which the flow-path responses Ui(t) depend on the flow parameters. In equation

(21), the left hand side represents the watershed unit response, while the righthand side is the sum of the flow-path responses corrected by a factor that accountsfor the cell area and for its tendency to generate excess precipitation. The values

of the flow parameters are tuned until equation (21) is satisfied.

The estimated excess precipitation and flow parameters can then be extrapolated

to other areas, if the same hydrologic behavior is assumed. This assumption,though, might be questionable when the areas used for calibration and application

are dissimilar. The values can also be used to estimate flow hydrographs at other

locations within the watershed, where hydrologic dissimilarity is less likely tooccur.

4. Case study: Waller Creek in Austin, Texas

Waller Creek is a 14.8 Km2 (3662 acres) watershed located within the urban coreof the City of Austin, Texas. Two flow gauging stations, set up by the US

Geological Survey (USGS), are located at 23rd and 38th Streets and have drainage

areas of 10.7 Km2 (2,643 acres) and 5.7 Km2 (1,416 acres) respectively. Themodel was calibrated with flow records of the station at 23rd Street and applied to



Waller Creek at 38th Street for comparison with observed flows. The period of

analysis ranged from October 14, 1994, 7:45 p.m. to October 17, 1994, 6:45 p.m..

A time step of 15 minutes was used and a total of 284 time intervals wereconsidered.

The watershed was delineated using a 30 m digital elevation model (DEM), andcomprised approximately 16,500 grid-cells. A map of the drainage-area of the two

flow-gauging stations and of the spatial distribution of the runoff coefficient is presented in Figure 5. It can be noted that just upstream of 38th Street there is a

large low-developed area that generates little runoff; while just upstream of 23rd

Street the area is more developed, yielding much more excess precipitation.



Figure 5: Waller Creek Watershed: Drainage area of the flow gauging stations

at 23rd and 38th Streets, creeks, and runoff coefficient distribution. Note that

only the gray areas generated runoff for this time period.

Additionally, from the flow records, it was noticed that the flow peaked first at

23rd Street and approximately 30 minutes later at 38th Street, which goes againstintuition because the peak time did not increase with drainage area. After the

direct-runoff/base-flow separation, it was found that, for this time period, 88% of the flow was direct runoff and 12% base flow. Note that, because of the high

impervious cover of the urban areas, direct runoff tends to be much more

important than base flow during storm events.

The methodology presented above was applied in the following steps:

STEP 1: The plot - in semi-logarithmic scale - of the flow record of the 23rd Street

station showed almost instantaneous peak times and long, straight and parallel

recession curves (see Figure 6), which suggests that the watershed responded toall storm events of the period with the same unit response function. This fact

confirms that a linear approximation is satisfactory for this hydrologic system,

because the response does not change from event to event.



STEP 2: To calculate the flow parameters (flow velocity and dispersion

coefficients), the watershed unit response obtained in STEP 1 was reproduced asthe aggregation of the flow-path responses according to equation (21). In order to

decrease the number of parameters (one flow velocity and one dispersion

coefficient per cell), it was assumed that water flows only as overland flow or stream flow, and a single velocity value was assigned to each type of flow; as

well, a uniform dispersion coefficient was taken for the entire watershed. Finally,

an abstractions coefficient equal to ζ = 0.4 was assumed. From the physical

viewpoint, this implies that all cells with runoff coefficient less than 0.4 generateno surplus. The value of ζ = 0.4 was chosen because most cells have runoff

coefficient much greater or much less than this value, and by selecting thisnumber it was presupposed that only highly developed areas generated runoff,

while lowly developed areas generated no runoff. After these assumptions, the

number of model parameters was reduced to three: (1) overland flow velocity, (2)

stream flow velocity, and (3) dispersion coefficient. By running an optimizationroutine, it was found that the flow parameters that produced the best match



between the observed watershed response and the one obtained as the aggregation

of the flow-path responses were: overland flow velocity = 0.27 m/s (0.898 fps), stream flow velocity = 0.27 m/s (0.898 fps) and dispersion coefficient = 1,629m2/s (17,535 ft2/s) (see Figure 7). Note that the optimization routine determined

the same value for both velocities, which indicates that in urban areas water flows

as fast over impermeable areas as it does in streams.

STEP 3: The watershed unit response, based on the parameters just calculated,

was then used to estimate the excess precipitation by deconvolution. Note that thecalculated excess precipitation and the observed flow follow the same trend,although the excess precipitation consists of somewhat concentrated pulses, while

the flow exhibits long recession curves following short rising limbs (see Figure

8).



STEP 4: Predicted flow in Waller Creek at 38th Street was determined using the

excess precipitation calculated in STEP 3, and the model parameters obtained inSTEP 2. Figure 9 presents observed flow (labeled Observed ) and predicted flow

at 38th Street. Three predicted flow series are plotted in this figure: the first one

(labeled No abstractions) assumes an abstractions parameter ζ = 0, i.e., cellcontributions proportional to the runoff coefficient; the second one (labeled

Abstractions = 0.4) assumes an abstractions parameter ζ = 0.4, i.e., cell

contributions proportional to the runoff coefficient minus 0.4; and the third one

(labeled Proportion) is obtained as the flow at 23rd Street multiplied by the ratioof the two drainage areas. It was interesting to notice that, at least in this case, the No abstractions series and the Proportion series were almost identical, the

difference being negligible for practical purposes. With regard to the No

abstractions series, it was observed that: (1) predicted values were consistently

higher than observed values, yielding a predicted flow volume that was 41%

greater than the observed volume; and (2) predicted values followed the trend of the observed values, but shifted approximately 30 minutes (2 time-steps) to the



left. To a great extent, these problems were solved in the Abstractions = 0.4

series. In this case: (1) the flow volume error went down to 4%, (2) the peak times

matched and no time-shift was observed, and (3) the recession curves werereproduced well.

The fact that the flow at 38th Street is only 39% of the flow at 23rd Street, instead

of 53% as the ratio of the areas, and the fact that the flow peaks first at 23rd Street

and 30 minutes later at 38th can be explained in the following way: (1) 38th Streetis fed by less developed areas than 23rd Street; (2) as an average, the developedareas that fed 38th Street are farther from the station than those that fed 23rd. This

explanation matches the geography of the area, and accounts for the peak shift

and runoff volume error. This type of hydrologic behavior evidences the need of accounting for the spatially variability of the hydrologic system.

5. Conclusions



It is known that the spatial variability of the terrain strongly affects storm runoff

processes. While representing a watershed by a small number of lumped

parameters (i.e. drainage area, channel slope, time of concentration) has theadvantage of simplifying the hydrologic modeling, it might miss some specific

local processes that affect the overall response of the system and that can not be

considered by lumped models.

Attempting to account for spatial variability of the terrain in storm runoff modeling, researchers have taken either of the following paths: (1) partitioning the

hydrologic system into subsystems and applying lumped models to each of them,

or (2) developing GIS user-interfaces to generate input files for, and display theresults of, other lumped models. In both cases, an improvement with respect to the

traditional fully lumped models has been accomplished, but these kind of

solutions can not be considered spatially distributed. In this article, the use of GISas a modeling tool itself, and not only as a link between the heterogeneous terrain

and an existing non-GIS lumped model, has been presented.

The model developed here is a generalization of the unit hydrograph model. This

generalized version of the unit hydrograph is used to route water in the landscape, provided that the hydrologic system is assumed to be linear. The model also

allows the user to consider time- and space-varying rainfall, thus relaxing some of

the basic assumptions of the unit hydrograph. The assumption of linearity, though,has not been relaxed by using this approach.

GIS appear to be an excellent environment for modeling spatially distributed

hydrologic processes, because they have spatial functions in the vector and raster

domain (some of them specifically developed for hydrologic purposes) and a

database management system, which combined allow one to perform hydrologicmodeling and calculations connected to geographic locations. In fact, GIS is able

to store and handle more spatially distributed terrain data than can be physically

obtained from the field. Thus, when dealing with distributed models, the problemis not necessarily how to develop GIS-based hydrologic models, or how to store

and handle the data, but how to get data in an amount that is consistent with the

model and hardware/software capabilities. At present, one of the limitations of this type of models is the scarcity of spatially distributed data. With regard to GIS

software, Arc/Info-Grid provides the necessary functions and commands to

analyze the digital elevation model (DEM) and obtain hydrologic features such aswatersheds, drainage areas, and flow lengths. It also provides the weighted flow

length function, FLOWLENGTH, that has been used to calculate the first and

second moments of the cell-outlet responses automatically, thus performingsequences of convolution integrals - in an approximated way - within the

Arc/Info-Grid environment.

The importance of accounting for spatial variability of the terrain when modeling

storm runoff was evidenced by the case of Waller Creek in Austin, Texas. In thiswatershed, and according to the data set used here, peak-time did not increase



with drainage area, and it was observed that the flow peaked first at 23rd Street

and 30 minutes later did so upstream at 38th Street. This situation reflected that

not all the terrain generates the same amount of runoff, and that the relativelocation of impervious areas in urban watersheds should be considered when

attempting to predict flows.

Predicted flow for Waller Creek at 38th Street matched reasonable well observed

records. Total runoff volume, peak times and recession curves were reproducedwell, and although some peak flow values were not matched, the overall tendency

was reproduced. This implies that not only the correct amount of water was

routed, but also at the correct velocity and with the correct spreading tendency.

Although definite conclusions could be drawn only after extensive testing of themethodology, it has become clear that the routing model is an improvement on the

currently used unit hydrograph, and that the handling of spatially distributed data

by Arc/Info proved to be adequate. However, a more elaborated excess

precipitation model might be necessary, especially if the excess precipitationcalculated at one watershed is to be exported to other watersheds.

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