Unit Hydrograph Analysis

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Jorge A. Ramírez
Water Resources, Hydrologic and Environmental Sciences Civil Engineering Department Colorado State University Fort Collins, Colorado 80523-1372, USA
ABSTRACT. The basic principles underlying the most commonly used physically-based models of the rainfall-runoff transformation process are reviewed. A thorough knowledge of these principles is a pre-requisite for flood hazard studies and, thus, this chapter reviews several physically-based methods to determine flood discharges, flow depths, and other flood characteristics. The chapter starts with a thorough review of linear system theory applied to the solution of hydrologic flood routing problems in a spatially aggregated manner -- Unit Hydrograph approaches. The chapter then proceeds to a review of distributed flood routing approaches, in particular the kinematic wave and dynamic wave approaches. The chapter concludes with a brief discussion about distributed watershed models, including single event models in which flow characteristics are estimated only during the flood, and continuous event models in which flow characteristics are determined continuously during wet periods and dry periods.
Flood prediction and modeling refer to the processes of transformation of rainfall into a flood hydrograph and to the translation of that hydrograph throughout a watershed or any other hydrologic system. Flood prediction and modeling generally involve approximate descriptions of the rainfall-runoff transformation processes. These descriptions are based on either empirical, or physically-based, or combined conceptualphysically-based descriptions of the physical processes involved. Although, in general, the conceptualizations may neglect or simplify some of the underlying hydrologic transport processes, the resulting models are quite useful in practice because they are simple and provide adequate estimates of flood hydrographs.
In modeling single floods, the effects of evapotranspiration, as well as the interaction between the aquifer and the streams, are ignored. Evapotranspiration may be ignored because its magnitude during the time period in which the flood develops is negligible when compared to other fluxes such as infiltration. Likewise, the effect of the stream-aquifer interaction is generally ignored because the response time of the subsurface soil system is much longer than the response time of the surface or direct runoff process. In addition, effects of other hydrologic processes such as interception and depression storage are also neglected. Event-based modeling generally involves the following aspects:
1Ramírez, J. A., 2000: Prediction and Modeling of Flood Hydrology and Hydraulics. Chapter 11 of Inland Flood Hazards: Human, Riparian and Aquatic Communities Eds. Ellen Wohl; Cambridge University Press.

Jorge A. Ramírez
a) evaluation of the rainfall flux over the watershed I(x, t) as a function of space and time;
b) evaluation of the rainfall excess or effective rainfall flux as a function of space and time, Ie(x, t). Effective rainfall is the rainfall available for runoff after infiltration and other abstractions have been accounted for; and
c) routing of the rainfall excess to the watershed outlet in order to determine the corresponding flood hydrograph, Q(t).
Hydrologic flood prediction models may be categorized into physical models and mathematical models. Mathematical models describe the system behavior in terms of mathematical equations representing the relationships between system state, input and output. Mathematical models can, in turn, be categorized as either purely conceptual models or physically-based models. Depending on whether the functions relating input, output and system state are functions of space and time, these models may be further categorized as lumped models or distributed models. Lumped models do not account explicitly for the spatial variability of hydrologic processes, whereas distributed models do. Lumped models use averages to represent spatially distributed function and properties.
Sherman (1932) first proposed the unit hydrograph concept. The Unit Hydrograph (UH) of a watershed is defined as the direct runoff hydrograph resulting from a unit volume of excess rainfall of constant intensity and uniformly distributed over the drainage area. The duration of the unit volume of excess or effective rainfall, sometimes referred to as the effective duration, defines and labels the particular unit hydrograph. The unit volume is usually considered to be associated with 1 cm (1 inch) of effective rainfall distributed uniformly over the basin area.
The fundamental assumptions implicit in the use of unit hydrographs for modeling hydrologic systems are:
a) Watersheds respond as linear systems. On the one hand, this implies that the proportionality principle applies so that effective rainfall intensities (volumes) of different magnitude produce watershed responses that are scaled accordingly. On the other hand, it implies that the superposition principle applies so that responses of several different storms can be superimposed to obtain the composite response of the catchment.
b) The effective rainfall intensity is uniformly distributed over the entire river basin.
c) The rainfall excess is of constant intensity throughout the rainfall duration.
d) The duration of the direct runoff hydrograph, that is, its time base, is independent of the effective rainfall intensity and depends only on the effective rainfall duration.

Jorge A. Ramírez

When the effective rainfall is given as a hyetograph, that is, as a sequence of M rainfall pulses of the same duration ∆t, the corresponding direct runoff hydrograph can be expressed as the discrete convolution of the rainfall hyetograph and the Unit Hydrograph as,


Qn = PmUn −m+ 1 , m* = min(n, M)


m= 1

Qn = Q(n∆t ),

n = 1,L, N


m ∆t

∫ Pm = Ie(τ )dτ ,

m = 1,L, M


(m −1)∆t

where Pm is the volume of the mth effective rainfall pulse, Qn is the direct runoff, and Unm+1 are the Unit Hydrograph ordinates.

Although the above assumptions lead to acceptable results, watersheds are indeed nonlinear systems. For example, unit hydrographs derived from different rainfall-runoff events, under the assumption of linearity, are usually different, thereby invalidating the linearity assumption.

The determination of unit hydrographs for particular basins can be carried out either using the theoretical developments of linear system theory; or using empirical techniques. For either case, simultaneous observations of both precipitation and streamflow must be available. These two approaches are presented in more detail in later sections.
Hydrograph Components

Total streamflow during a precipitation event includes the baseflow existing in the basin prior to the storm and the runoff due to the given storm precipitation. Total streamflow hydrographs are usually conceptualized as being composed of:

a) Direct Runoff, which is composed of contributions from surface runoff and quick interflow. Unit hydrograph analysis refers only to direct runoff.

b) Baseflow, which is composed of contributions from delayed interflow and groundwater runoff.

Surface runoff includes all overland flow as well as all precipitation falling directly onto stream channels. Surface runoff is the main contributor to the peak discharge.

Interflow is the portion of the streamflow contributed by infiltrated water that moves laterally in the subsurface until it reaches a channel. Interflow is a slower process


Jorge A. Ramírez
than surface runoff. Components of interflow are quick interflow, which contributes to direct runoff, and delayed interflow, which contributes to baseflow (e.g., Chow, 1964.)
Groundwater runoff is the flow component contributed to the channel by groundwater. This process is extremely slow as compared to surface runoff.
Schematically in Figure 11.1, the streamflow hydrograph is subdivided into a) Rising Limb: rising portion of the hydrograph, composed mostly of surface runoff. b) Crest: zone of the hydrograph around peak discharge. c) Falling (or Recession) Limb: Portion of the hydrograph after the peak discharge, composed mostly of water released from storage in the basin. The lower part of this recession corresponds to groundwater flow contributions.
The main factors affecting hydrograph shape are:
1) Drainage characteristics: basin area, basin shape, basin slope, soil type and land use, drainage density, and drainage network topology. Most changes in land use tend to increase the amount of runoff for a given storm (e.g., Chow et al., 1988; Singh, 1989; Bras, .1990).
2) Rainfall characteristics: rainfall intensity, duration, and their spatial and temporal distribution; and storm motion, as storms moving in the general downstream direction tend to produce larger peak flows than storms moving upstream (e.g., Chow et al., 1988; Singh, 1989; Bras, .1990).
Hydrographs are also described in terms of the following time characteristics (see Figure 11.1):
Time to Peak, tp: Time from the beginning of the rising limb to the occurrence of the peak discharge.
The time to peak is largely determined by drainage characteristics such as drainage density, slope, channel roughness, and soil infiltration characteristics. Rainfall distribution in space also affects the time to peak.
Time of Concentration, tc: Time required for water to travel from the most hydraulically remote point in the basin to the basin outlet. For rainfall events of very long duration, the time of concentration is associated with the time required for the system to achieve the maximum or equilibrium discharge. Kibler (1982) and Chow et al. (1988) summarize several of many empirical and physically-based equations for tc that have been developed.
The drainage characteristics of length and slope, together with the hydraulic characteristics of the flow paths, determine the time of concentration.
Lag Time, tl: Time between the center of mass of the effective rainfall hyetograph and the center of mass of the direct runoff hydrograph.

Jorge A. Ramírez

The basin lag is an important concept in linear modeling of basin response. The lag time is a parameter that appears often in theoretical and conceptual models of basin behavior. However, it is sometimes difficult to measure in real world situations. Many empirical equations have been proposed in the literature. The simplest of these equations computes the basin lag as a power function of the basin area.
Time Base, tb: Duration of the direct runoff hydrograph.
Baseflow separation.
As the Unit Hydrograph concept applies only to direct runoff, the direct runoff must be separated from the baseflow. Baseflow separation or hydrograph analysis is the process of separating the direct runoff (surface runoff and quick interflow) from the baseflow. This separation is somewhat arbitrary, but corresponds to theoretical concepts of basin response.
Subjective methods. Several subjective methods are shown in Figure 11.1. The simplest one consists in arbitrarily selecting the discharge marking the beginning of the rising limb as the value of the baseflow and assuming that this baseflow discharge remains constant throughout the storm duration. A second method consists in arbitrarily selecting the beginning of the groundwater recession on the falling limb of the hydrograph (usually assumed to occur at a theoretical inflection point) and connecting this point by a straight line to the beginning of the rising limb. A third example of subjective methods consists in extending the recession prior to the storm by a line from the beginning of the rising limb to a point directly beneath the peak discharge and then connecting this point to the beginning of the groundwater recession on the falling limb.
Area method. The area method of baseflow separation consists in determining the beginning of the baseflow on the falling limb with the following empirical equation,

N = bA0.2


relating the time in days from the peak discharge, N, to the basin area, A. When A is in square miles, b equals 1. When A is in square kilometers, b equals 0.8. This equation is unsuitable for smaller watersheds and should be checked for a number of hydrographs before using.

The master recession curve method. This method consists in modeling the response of the groundwater aquifer as a linear reservoir of parameter k. This assumption leads to the following equation for the groundwater recession hydrograph,



Q(t )e−(t −t o ) o




where Q(t) is the baseflow at time t; Q(to) is a reference baseflow discharge at time to, and k is the recession constant for baseflow. This method is based on a linear reservoir model of unforced basin response (that is, response from storage) and it can be used to separate


Jorge A. Ramírez
the contributions to the recession flow from surface storage, subsurface storage, and groundwater aquifer storage. It involves the determination of several recession constants.
Effective precipitation Streamflow hydrograph Baseflow hydrographs
tl tp
tb Figure 11.1: Schematic Description of Hydrograph UNIT HYDROGRAPHS: EMPIRICAL DERIVATION The following are essential steps in deriving a unit hydrograph from a single storm: 1) Separate the baseflow and obtain the direct runoff hydrograph (DRH). 2) Compute the total volume of direct runoff and convert this volume into equivalent depth of effective rainfall (in centimeters or in inches) over the entire basin. 3) Normalize the direct runoff hydrograph by dividing each ordinate by the equivalent volume (in or cm) of direct runoff (or effective rainfall).

Jorge A. Ramírez
4) Determine the effective duration of excess rainfall. To do this, obtain the effective rainfall hyetograph (e.g., use the φ-index, the Horton, Green and Ampt, or Philip equations, or some other method to determine infiltration losses) and its associated duration. This duration is the duration associated with the unit hydrograph.
Unit hydrographs are fundamentally linked to the duration of the effective rainfall event producing them. They can only be used to predict direct runoff from storms of the same duration as that associated with the UH, or from storms which can be described as a sequence of pulses, each of the same duration as that associated with the UH (see Equation 1).
Unit Hydrographs for Different Effective Duration
A unit hydrograph for a particular watershed is developed for a specific duration of effective rainfall. When dealing with a rainfall of different duration a new unit hydrograph must be derived for the new duration. The linearity property implicit in the UH analysis can be used to generate UH’s associated with larger or smaller effective rainfall pulse duration. This procedure is sometimes referred to as the S-curve Hydrograph method.
S-Curve Hydrograph Method
An S-hydrograph represents the response of the basin to an effective rainfall event of infinite duration. Assume that an UH of duration D is known and that an UH for the same basin but of duration D’ is desired. The first step is to determine the S-curve hydrograph by adding a series of (known) UH’s of duration D, each lagged by a time interval D. The resulting superposition represents the runoff resulting from a continuous rainfall excess of intensity 1/D.
Lagging the S-curve in time by an amount D’ and subtracting its ordinates from the original unmodified S-curve yields a hydrograph corresponding to a rainfall event of intensity 1/D and of duration D’. Consequently, to convert this hydrograph whose volume is D’/D into a unit hydrograph of duration D’, its ordinates must be normalized by multiplying them by D/D’. The resulting ordinates represent a unit hydrograph associated with an effective rainfall of duration D’.
A hydrologic system (a basin) is said to be a linear system if the relationship between storage, inflow, and outflow is such that it leads to a linear differential equation. The hydrologic response of such systems can be expressed in terms of an impulse response function (IRF) through a so-called Convolution Equation. Linear systems possess the properties of additivity and proportionality, which are implicit in the convolution equation. Linear reservoirs, for example, are special cases of a general hydrologic system model in which the storage is linearly related to the output by a constant k.
Impulse Response Function

Jorge A. Ramírez

The IRF of a linear system represents the response of the system to an instantaneous impulse (excitation) of unit volume applied at the origin in time (t=0). The response of continuous linear systems can be expressed, in the time domain, in terms of the impulse response function via the convolution integral as follows,


Q(t) = ∫ Ie (τ )u(t − τ )dτ



where u(t) is the impulse response function of the system.

In hydrology, it has been customary to assume that watersheds behave as linear systems. When dealing with hydrologic systems, u(t) represents the instantaneous unit hydrograph (IUH), and Q(t) and Ie(t) represent direct runoff and excess or effective precipitation, respectively. Thus, an Instantaneous Unit Hydrograph represents the response of a watershed (discharge at its outlet as a function of time) to a unit volume of precipitation uniformly distributed over the basin and occurring instantaneously at time t = 0.
Unit Step Response Function

The unit step response function (SRF) is the theoretical counterpart to the S-curve hydrograph concept presented earlier in the empirical UH analysis section. It represents the runoff hydrograph from a continuous effective rainfall of unit intensity. As can be seen from its definition, it is the convolution of 1 and u(t), and obtained as,


g(t) = ∫ u(t)dt



Unit Pulse Response Function

The unit pulse response function (PRF) is the theoretical counterpart to the UH concept presented earlier. It represents the runoff hydrograph from a constant effective rainfall of intensity 1/∆t and of duration ∆t.

∫ h(t) = 1 [g(t) − g(t − ∆t)] = 1 t u(τ )dτ (6)


∆t t −∆t

From its definition, the PRF can be seen as the normalized difference between two lagged SRF’s (S-curve hydrographs), lagged by an amount ∆t. This is analogous to the procedure presented earlier in the section on S-hydrograph analysis.
Discrete Convolution Equation

When the effective rainfall is given as a hyetograph, that is, as a sequence of rainfall pulses of the same duration ∆t, the corresponding direct runoff hydrograph can be expressed as the discrete convolution of the rainfall hyetograph and a Unit Hydrograph,


Jorge A. Ramírez


Qn = PmUn −m+ 1 , m* = min( n, M )


m= 1

Qn = Q(n∆t ),

n = 1,L, N


m ∆t

∫ Pm = Ie(τ )dτ ,

m = 1,L, M


(m −1)∆t

where Pm is the volume of the mth effective rainfall pulse and the Unit Hydrograph ordinates are given by,

∫ 1 (n−m+1)∆t

U n−m+1 = h[(n − m + 1)∆t] =

u(τ )dτ


∆t (n−m)∆t

The UH ordinates correspond to the area under the IUH between two consecutive time intervals.

The discrete convolution can be expressed alternatively as a matrix equation as,

[P][U ] = [Q]


 P1

 Q1 

 P2 P1

 Q2 

 P2 L  U1   

 P


P 


 =

 Q



1 


 

 PM

P2 U   


N − M +1

 

PM 

QN 

where M is the number of effective rainfall pulses; N-M+1 is the number of UH ordinates; and N is the number of direct runoff ordinates. Matrix P is of dimensions (N x (N-M+1)). Recasting the discrete convolution equation in this manner allows for an objective (and optimal) mathematical derivation of the UH ordinates. The estimation of the Unit Hydrograph from simultaneous observations of effective precipitation (Pm) and direct runoff (Qn) can be seen as a linear static estimation problem for which a method like least-squares, linear programming, or others can be used.

The least-squares approach tries to obtain a set of UH ordinates that minimizes the sum of squares of the errors, and leads to the following solution for the UH,

[U ] = [[ P]T [ P]]−1 [ P]T [ Q]



Jorge A. Ramírez

where [[P]T[P]]-1 is the inverse of matrix [[P]T[P]], and the superscript T stands for transpose. This approach is adequate when the data (precipitation and discharge) are relatively error free or the errors can be expected to be small. When the data can be expected to contain large errors a different approach may be more adequate. Such approach would minimize the sum of the absolute value of the errors. Linear programming would then be the appropriate solution procedure.


As indicated earlier, sets of concurrent observations of effective rainfall and direct runoff are required for the derivation of unit hydrographs. Thus, the resultant UH is specific to the particular watershed defined by the point on the stream where the direct runoff observations were made. When no direct observations are available, or when UH’s for other locations on the stream in the same watershed or for nearby watersheds of similar characteristics are required, Synthetic Unit Hydrograph procedures must be used.

Synthetic Unit Hydrograph procedures can be categorized as (e.g., Chow et al., 1988): 1) those based on models of watershed storage (e.g., Nash, 1957, 1958, 1959; Dooge, 1959; etc.); 2) those relating hydrograph characteristics (time to peak, peak flow, etc.) to watershed characteristics (e.g., Snyder, 1938; Geomorphologic Instantaneous Unit Hydrograph); and 3) those based on a dimensionless unit hydrograph (e.g., Soil Conservation Service, 1972).
Conceptual UH’s Based on Models of Watershed Storage

The Nash and Dooge Models.

Linear Reservoirs - A linear reservoir is characterized by a linear relationship between the storage and the output as,

S(t) = kQ(t)


The impulse response function of such linear reservoir is,

u(t ) = 1 e−t / k



Nash (1957,1958, 1959, 1960) proposed a cascade of n equal linear reservoirs as a model on which to base the derivation of IUH’s for natural watersheds. The Nash model is one of the most widely used models in applied hydrology. Using the convolution equation (eq. 4) and the impulse response function for a single linear reservoir (eq. 11), the IUH corresponding to the Nash Model can be easily obtained as follows,
un (t) = kΓ1(n) (kt )n−1 e−t/ k (12)


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Unit Hydrograph Analysis