APPLICATION OF THE GENERAL DIMENSIONLESS UNIT HYDROGRAPH
USING CALIFORNIA WATERSHED DATA
Luis Gustavo Ariza Trelles
The concept of unit hydrograph is well established in hydrologic engineering research and practice. The unit hydrograph is defined as the hydrograph produced by a unit depth of runoff uniformly distributed over the entire catchment (watershed or basin) and lasting a specified unit duration. The concept has been used since the 1930s for the simulation of flood flows around the world (Sherman, 1932). The general dimensionless unit hydrograph (GDUH), developed by Ponce (2009a, 2009b), is a dimensionless formulation of the unit hydrograph. The GDUH effectively associates the convolution technique (of the unit hydrograph) with the model of cascade of linear reservoirs (CLR), originally due to Nash (1957). The CLR model constitutes the routing component of several hydrologic models that have since been developed around the world, notably the SSARR model (U.S. Army Engineer North Pacific Division, 1972).
This study attempts to validate the GDUH model using California watershed/basin data. Geographic and rainfall-runoff
data is readily
available online in the State of California, thus facilitaling collection and analysis. Digital elevation maps (DEM)
are available from the USGS virtual platforms
.
Rainfall data is available from the NOAA virtual platform
Alaska Satellite Facility. Runoff data
is available from the USGS virtual platform
National Centers for Environmental Information.
National Water Information SystemThis study selects ten (10) California watersheds/basins for analysis. To enable the proper study of unit hydrograph diffusion, the basins encompass a wide range in the values of geomorphological parameters (drainage area, average land surface slope, and stream channel slope). Conceptual and statistical analyses are used to develop a methodology for the accurate prediction of unit hydrographs on the basis of local/regional geomorphology. Given the prospect of global warming and its magnifying effect on flood flows, the timeliness of this endeavor cannot be overemphasized.
The objectives of this study are:
To validate and test the general dimensionless unit hydrograph (GDUH) model using suitable geomorphological parameters.
To select ten (10) watersheds/basins in California for detailed conceptual and statistical analysis. For each basin: To produce a set of maps and calculate related suitable geomorphological parameters using GIS. To calculate *average measured*and*predicted*unit hydrographs following the GDUH methodology.
To identify a diffusion parameter to better characterize unit hydrograph diffusion. To use statistical correlations to develop a predictive tool for unit hydrograph analysis based on local/regional geomorphology.
This study encompasses the development and validation of a predictive methodology to calculate unit hydrographs based on local/regional geomorphology. The avowed strength of the methodology is its conceptual basis, being based on time-tested cascade of linear reservoirs theory. The central focus on the general dimensionless unit hydrograph (GDUH) as a unifying theory enhances the validation exercise.
Over the past century, the unit hydrograph (UH) has been used as a methodology to generate flood
flows for midsize and large basins
(Ponce, 1989;
Ponce, 2014a).
In 1930, the
This statement may be interpreted as follows: For a certain basin of drainage area V and consequently the peak flow _{r}Q, are proportional
to the effective rainfall intensity _{p}d/. In other words, the hydrograph response (t_{r}Q)
is linear with respect to the intensity and, therefore, independent of the time base T.
_{b}
Sherman (1932) built on this
concept to develop the unit hydrograph for flood studies in large basins.
The word
The unit hydrograph is defined as the hydrograph produced by a
Two assumptions are crucial to the development of the unit hydrograph: (1)
The time base of all hydrographs obtained in this way is equal to that of the unit hydrograph. Therefore,
the procedure can be used to calculate hydrographs produced by a storm consisting of a series
of runoff depths, each
The summation of the corresponding ordinates of these hydrographs (superposition) allows the calculation of
the composite hydrograph, as shown in Fig. 2.2 (c). The procedure depicted in Fig. 2.2 is
referred to as the
The assumption of linearity has long been considered one of the limitations of unit hydrograph theory. In nature, it is unlikely that catchment response will always follow a linear function. For one thing, discharge and mean velocity are nonlinear functions of flow depth and stage. In practice, however, the linear assumption provides a convenient means of calculating runoff response without the complexities associated with nonlinear analysis.
The upper limit of applicability of the unit hydrograph is not very well defined. Sherman (1932) used it in
connection with basins varying from 1300 to 8000 km
As shown in Section 2.3, the concepts of unit hydrograph and The techniques for storage routing are invariably based on the differential equation of water storage. This equation is founded on the principle of mass conservation, which states that the change in flow per unit length in a control volume is balanced by the change in flow area per unit time. In partial differential form it is expressed as follows:
in which The differential equation of storage is obtained by lumping spatial variations. For this purpose, Eq. 2-1 is expressed in finite increments:
With Δ
in which inflow, outflow, and rate of change of storage are expressed in
Equation 2-4 implies that any difference between inflow and outflow is balanced by a change of storage in time (Fig. 2.3). In a typical reservoir routing application, the inflow hydrograph (upstream boundary condition), initial outflow and storage (initial conditions), and reservoir physical and operational characteristics are known. Thus, the objective is to calculate the outflow hydrograph for the given initial condition, upstream boundary condition, reservoir characteristics, and operational rules.
Equation 2-4 can be solved by analytical or numerical means.
The numerical approach is usually preferred because it can account for an arbitrary inflow hydrograph.
The solution is accomplished by discretizing Eq. 2-4 on the
Figure 2.4 shows two consecutive time levels, 1 and 2, separated between them an interval Δ
in which
For linear reservoirs, the relation between storage and outflow is linear. Therefore:
and
in which
Substituting Eqs. 2-6 into 2-5, and solving for
in which
Since
A reservoir exerts a diffusive action on the flow, with the net result that the peak flow is attenuated and
consequently, the time base is increased.
For the case of a linear reservoir, the amount of attenuation is a function of
The cascade of linear reservoirs is a widely used method of hydrologic catchment routing.
As its name implies, the method is based on the connection of several linear reservoirs in series.
For
Each reservoir in the series provides a certain amount of diffusion and associated lag.
For a given set of parameters Δ
The solution of the cascade of linear reservoirs can be accomplished in two ways: (1) analytical, and (2) numerical.
The analytical version is due to t ⇒ 0.
Nash assumed that the IUH could be represented as the cascade of linear reservoirs.
_{r}The numerical version of the cascade of linear reservoirs is featured in several hydrologic simulation models developed in the United States and other countries. Notable among them is the Streamflow Synthesis and Reservoir Regulation (SSARR) model, which uses it in its watershed, stream channel routing, and baseflow modules. The SSARR model has been in the process of development and application since 1956. The model was developed to meet the needs of the U.S. Army Corps of Engineers North Pacific Division in the area of mathematical hydrologic simulation for planning, design, and operation of water-control works. (U.S. Army Engineer North Pacific Division, 1972). The SSARR model was first applied to operational flow forecasting and river management activities in the Columbia River System. Later, it was used by U.S. Army Corps of Engineers, National Weather Service, and Bonneville Power Administration. Numerous river systems in the United States and other countries have been modeled with SSARR. To derive the routing equation for the method of cascade of linear reservoirs, Eq. 2-7 is reproduced here in a slightly different form:
in which
As with Eq. 2-7, the routing coefficients
For application to catchment routing, it is convenient to define the average inflow as follows:
Substituting Eqs. 2-10b and 2-11 into Eq. 2-9 gives the following:
or, alternatively, through some algebraic manipulation:
Equations 2-12 and 2-13 are in a form convenient for catchment routing because the inflow is usually a rainfall hyetograph, that is, a constant average value per time interval. Note that Eqs. 2-12 and 2-13 are identical. Equation 2-12 was presented by Ponce in his version of the cascade of linear reservoirs (Ponce, 2014a). Equation 2-13 is the routing equation of the SSARR model (U.S. Army Engineer North Pacific Division, 1972).
Smaller values of The cascade of linear reservoirs provides a convenient mechanism for simulating a wide range of catchment routing problems. Furthermore, the method can be applied to each runoff component (surface runoff, subsurface runoff, and baseflow) separately, and the catchment response can be taken as the sum of the responses of the individual components.
For instance, assume that a certain basin has 10 cm of runoff, of which 7 cm are surface runoff, 2 cm are subsurface runoff, and 1 cm is baseflow.
Since surface runoff is the less diffused process, it can be simulated with a high Courant number, say
According to Nash, the general equation for the instantaneous unit hydrograph is:
in which Equation 2-14 is the analytical version of the IUH or cascade of linear reservoirs. The numerical version is represented by either Ponce's model (Eq. 2-12) or the SSARR model (Eq. 2-13).
Rodríguez-Iturbe and Valdés (1979) pioneered in establishing the relation
of the instantaneous unit hydrograph with the geomorphologic characteristics of the catchment;
see also the companion papers (Valdés The geomorphologic characteristics are expressed in terms of the following basin parameters: The *bifurcation ratio*(law of stream numbers)*R*:_{B}*R*_{B}=*N*_{w}/*N*_{w+1}(2-15a) In any basin, *N*_{w}is the number of streams of order*w*, and*N*_{w+1}is the number of streams of order*w*+1. In Nature, values of*R*_{B}are normally between 3 to 5.The *length ratio*(law of stream lengths)*R*:_{L}*R*_{L}=*L̄*_{w}/*L̄*_{w-1}(2-15b) In any basin, *L̄*_{w}is the mean hydraulic length of order*w*, and*L̄*_{w-1}is the mean hydraulic length of order*w*-1. In Nature, values of*R*_{L}are normally between 1.5 to 3.5.The *area ratio*(law of stream areas)*R*:_{A}*R*_{A}=*Ā*_{w}/*Ā*_{w-1}(2-15c) In any basin, *Ā*_{w}is the mean area of order*w*, and*Ā*_{w-1}is the mean area of order*w*-1. In Nature, values of*R*_{A}are normally between 3 to 6.The *internal scale parameter**L*_{Ω}, defined as the hydraulic length of a basin of order*Ω*.The *dynamic parameter v*, defined as the velocity corresponding to the peak discharge for a given rainfall-runoff event in a basin.
According to Rodríguez-Iturbe and Valdés (1979), the equations to calculate the geomorphologic instantaneous unit hydrograph (GIUH) are:
In which
The parameters
The parameters
Equations 2-18 and 2-19 assume the basin order
The unit hydrograph seeks to calculate runoff diffusion, i.e., the spreading of the hydrograph in time and space.
In practice, the amount of runoff diffusion depends on whether the flow is through: (a) a reservoir,
Flow through a reservoir
Reservoirs are natural or artificial surface-water hydraulic features that provide runoff diffusion. Runoff diffusion is depicted by the sizable attenuation of the inflow hydrograph, as shown in Fig. 2.6.
Stream channels, i.e., channels or canals, are surface-water hydraulic features which may or may not provide runoff diffusion,
depending on the relative scale of the disturbance (flood wave). The amount of wave diffusion is characterized by the
dimensionless wavenumber
in which
Four types of waves are identified: - Kinematic waves,
- Diffusion waves,
- Mixed kinematic-dynamic waves, and
- Dynamic waves.
Kinematic waves lie on the left side of the wavenumber spectrum, featuring constant dimensionless relative wave celerity and zero attenuation. Dynamic waves lie on the right side, featuring constant dimensionless relative wave celerity and zero attenuation. Mixed kinematic-dynamic waves lie in the middle of the spectrum, featuring variable dimensionless relative wave celerity and medium to high attenuation. Diffusion waves are intermediate between kinematic and mixed kinematic-dynamic waves, featuring mild attenuation. In hydraulic engineering practice, dynamic waves are commonly referred to as Lagrange or "short" waves, while the mixed kinematic-dynamic waves are commonly referred to as "dynamic waves," fueling a semantic confusion.
For flood routing computations, the governing equations of continuity and motion, commonly referred to as the
Saint Venant equations,
may be linearized and combined into a convection-diffusion equation
with discharge
in which
in which d = mean flow depth,
and _{o}g = gravitational acceleration.
In Eq. 2-22, for = 1, the coefficient of the second-order term reduces to zero,
and the diffusion term vanishes. Under this latter flow condition, all waves, regardless of scale,
travel with the same speed, fostering the development of roll waves (Fig. 2.8).
V
Surface runoff in catchments may be one of three types (Ponce, 1989a; 2014a): Concentrated flow, when the effective rainfall duration is equal to the time of concentration, Superconcentrated flow, when the effective rainfall duration is longer than the time of concentration, and Subconcentrated flow, when the effective rainfall duration is shorter than the time of concentration.
Figure 2.9 shows a typical open-book schematization for overland flow modeling. Input is effective rainfall on two planes adjacent ot a channel. Output is the outflow hydrograph at the catchment outlet.
Figure 2.10 shows dimensionless catchment outflow hydrographs for the three cases described above
(Ponce and Klabunde, 1999).
The maximum possible peak outflow is: I,
in which _{e} AI = effective rainfall intensity, and _{e}A = catchment area.
By definition, the maximum possible peak outflow is reached for
superconcentrated and concentrated flow.
However, in the case of subconcentrated flow, the peak outflow fails to reach the maximum possible value. Effectively, this
amounts to runoff diffusion, because the flow has actually been spread in time (and space).
Thus, runoff diffusion is produced for all waves when the time of concentration exceeds the effective rainfall duration. This is typically the case of midsize and large basins, for which the catchment slope (along the hydraulic length) is sufficiently mild (small). The time of concentration is directly related to catchment hydraulic length and bottom friction, and inversely related to bottom slope and effective rainfall intensity (Ponce, 1989b; 2014b).
Figure 2.11 shows dimensionless rising overland flow hydrographs for a kinematic wave model (labeled KW)
and for several storage-concept models,
for the discharge-area rating exponent
In actual numerical computations, a kinematic wave model may not be entirely devoid of diffusion, due to the appearance of numerical diffusion (Cunge, 1969; Ponce, 1991a). In fact, first-order schemes of the kinematic wave equation produce numerical diffusion. This diffusion, however, is uncontrolled, not based on physical parameters and, therefore, unrelated to the true diffusion of the physical problem.
The cascade of linear reservoirs (CLR) (Section 2.3) and the instantaneous unit hydrograph
(IUH) (Section 2.4) are essentially the same. A general dimensionless unit hydrograph (GDUH) may be generated using the
The dimensionless time
in which
The dimensionless discharge
in which
in which:
Therefore:
In SI units, for a unit rainfall depth of 1 cm:
Thus:
in which A in km^{2}.
In practice, a set of Once the GDUH is chosen, the ordinates of the unit hydrograph may be calculated from Eq. 2-25 as follows:
Likewise, the abscissa (time) may be calculated from Eq. 2-20 as follows:
The unit hydrograph thus calculated may be convoluted with the effective storm hyetograph to determine the composite flood hydrograph (Ponce, 2014). The GDUH has the following significant advantages: The GDUH is solely a function of *C*and*N*, and is of global applicability.Unlike other established unit hydrograph procedures such as the Natural Resources Conservation Service (NRCS) unit hydrograph, the GDUH is a two-parameter model; therefore, it is able to simulate a wider range of runoff diffusion effects (Ponce, 2014).
The GDUH cascade parameters (
In Nature, basins are classified with regards to runoff diffusion on the basis of mean land surface slope. A preliminary classification is shown in
The methodology for this study
aims to develop a relation between the GDUH cascade parameters The selected methodology depends on the temporal storm characteristics. The following two situations are considered: Simple storms, featuring a one-day precipitation impulse (a one-day predominant precipitation event may be used in practice); and Complex storms, with a precipitation event distributed over several days.
For simple storms, the following steps are required: *Assemble the rainfall-runoff data*Assemble corresponding sets of rainfall-runoff data for each watershed/basin, and identify three (3) suitable infrequent events for analysis. *Calculate the unit hydrograph runoff volume*Calculate the runoff volume corresponding to 1 cm of effective rainfall. *For each event:*Use the straight line technique of baseflow separation to determine the direct runoff storm hydrograph. Calculate the runoff volume corresponding to the direct runoff storm hydrograph obtained in Step 3(a), and compare with the runoff volume obtained in Step 2. Based on the results of Step 3(b), multiply the direct runoff storm hydrograph ordinates by the appropriate factor to establish the unit hydrograph ordinates. Confirm that it corresponds to 1 cm of runoff. When warranted, perform minor volumetric corrections. Calculate the dimensionless unit hydrograph (DUH) using Eqs. 2-20 and 2-23 for the abscissas and ordinates, respectively.
*Calculate the unit hydrograph*Average the three (3) dimensionless unit hydrographs obtained in Step 3 (d) to obtain the watershed/basin's dimensionless unit hydrograph (DUH). Confirm that it corresponds to 1 cm of runoff. *Calculate the cascade parameters C and N*Match the dimensionless unit hydrograph peak flow *Q*_{*}_{p}and time-to-peak*t*_{*}_{p}to a suitable general dimensionless unit hydrograph (GDUH) featuring paired cascade parameters .*C*and*N*
For complex storms, the follow steps are required: *Assemble the rainfall-runoff data*Assemble corresponding sets of rainfall-runoff data for each watershed/basin, and identify three (3) suitable infrequent events for analysis. *For each event:*Use the straight line technique of baseflow separation to determine the direct runoff storm hydrograph. Calculate the runoff volume corresponding to the direct runoff storm hydrograph. Apply the *φ*-index procedure to the total storm hyetograph to determine the effective storm hyetograph (Ponce, 2014a).Apply the *inverse convolution*technique to the direct runoff storm hydrograph obtained in Step 2 (b) and the effective storm hyetograph obtained in Step 2 (c) to calculate the unit hydrograph (Section 3.2).Calculate the dimensionless unit hydrograph (DUH) using Eqs. 2-20 and 2-23 for the abscissas and ordinates, respectively.
*Calculate the unit hydrograph*Average the three (3) dimensionless unit hydrographs obtained in Step 2 (e) to obtain the watershed/basin's unit hydrograph (UH). Confirm that it corresponds to 1 cm of runoff. *Calculate the cascade parameters C and N*Match the dimensionless unit hydrograph peak flow *Q*_{*}_{p}and time-to-peak*t*_{*}_{p}to a suitable general dimensionless unit hydrograph (GDUH) featuring paired cascade parameters .*C*and*N*
For each of the basins analyzed, the set of thus found paired
In a practical application, once the average stream channel slope and land surface slope are determined, the appropriate values of
Convolution is the procedure by which a certain unit hydrograph and an effective storm hyetograph are used to calculate the
corresponding flood hydrograph. Conversely,
Fig. 3.1 Convolution and inverse convolution. |