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5.1 MIDSIZE CATCHMENTS
A midsize catchment is described by the following features:
A catchment possessing some or all of the above properties is midsize in a hydrologic sense. Since rainfall intensity varies within the storm duration, catchment response is described by methods that take explicit account of the temporal variation of rainfall intensity. The most widely used method to accomplish this is the unit hydrograph technique. In an nutshell, it consists of deriving a hydrograph for a unit storm (the unit hydrograph) and using it as a building block to develop the hydrograph corresponding to the actual effective storm hyetograph.
In unit hydrograph analysis, the duration of the unit hydrograph is usually a fraction of the time of concentration. The increase in time of concentration is due to the larger drainage area and the associated reduction in overall catchment gradient. The assumption of uniform spatial distribution of rainfall is a characteristic of midsize catchment analysis. This assumption allows the use of a lumped method such as the unit hydrograph.
Unlike midsize catchments, for large catchments rainfall is likely to vary spatially, either as a general storm of concentric isohyetal distribution covering the entire catchment with moderate rainfall or as a highly intensive local storm (thunderstorm) covering only a portion of the catchment.
An important feature of large catchments that sets them apart from midsize catchments is their substantial capability for channel storage. Channel storage processes act to attenuate the flows while in transit in the river channels. Attenuation can be due either to longitudinal storage (for inbank flows) or to lateral storage (for overbank flows). In the first case, the storage amount is largely controlled by the slope of the main channel. For catchments with mild channel slopes, channel storage is substantial; conversely, for catchments with steep channel slopes, channel storage is negligible. Since large catchments are likely to have mild channel slopes, it follows that they have a substantial capability for channel storage.
In practice, this means that large catchments cannot be analyzed with spatially lumped methods such as the unit hydrograph, since these methods do not take explicit account of channel storage processes. Therefore, unlike for midsize catchments, for large catchments it may be necessary to use channel routing (Chapter 9) to account for the expanded role of river flow in the overall runoff response.
As with the limit between small and midsize catchments, the limit between midsize and large catchments is not immediately apparent. For midsize catchments, runoff response is primarily a function of the characteristics of the storm hyetograph, with time of concentration playing a secondary role. Therefore, the latter is not well suited as a descriptor of catchment scale. Values ranging from 100 to 5000 km2 have been variously used to define the limit between midsize and large catchments. While there is no consensus to date, the current trend is toward the lower limit. In practice, it is likely that there would be a range of sizes within which both midsize and large catchment techniques are applicable. However, the larger the catchment area, the less likely it is that the lumped approach is able to provide the necessary spatial detail.
It should be noted that the techniques for midsize and large catchments are indeed complementary. A large catchment may be viewed as a collection of midsize subcatchments. Unit hydrograph techniques can be used for subcatchment runoff generation, with channel routing used to connect streamflows in a typical dendritic network fashion (Fig. 5-1). An example of a hydrologic computer model using the network concept is the HEC-HMS model (Hydrologic Modeling System) of the U.S. Army Corps of Engineers. This and other computer models are described in Chapter 13.
In practice, channel-routing techniques are not necessarily restricted to large catchments. They can also be used for midsize catchments and even for small catchments. However, the routing approach is considerably more complicated than the unit hydrograph technique. The routing approach is applicable to cases where an increased level of detail is sought, above that which the unit hydrograph technique is able to provide; for instance, when the objective is to describe the temporal variation of streamflow at several points inside the catchment. In this case, the routing approach may well be the only way to accomplish the modeling objective.
The hydrologic description of midsize catchments consists of two processes:
This chapter focuses on a method of rainfall abstraction that is widely used for hydrologic design in the United States: the Natural Resources Conservation Service (NRCS) runoff curve number method. Other rainfall abstraction procedures used by existing computer models are discussed in Chapter 13.
With regard to hydrograph generation, this chapter centers on the unit hydrograph technique, which is a defacto standard for midsize catchments, having been used extensively throughout the world. The NRCS TR-55 method, also included in this chapter, has peak flow and hydrograph generation capabilities and is applicable to small and midsize urban catchments with time of concentration in the range 0.1-10 h. The TR-55 method is based on the runoff curve number method, unit hydrograph techniques, and simplified stream channel routing procedures.
5.2 RUNOFF CURVE NUMBER
The runoff curve number method is a procedure for hydrologic abstraction of storm rainfall developed by the U.S. Natural Resources Conservation Service (formerly Soil Conservation Service) . In this method, total storm runoff depth is a function of total storm rainfall depth and an abstraction parameter referred to as runoff curve number, curve number, or CN. The curve number varies in the range 1 ≤ CN ≤ 100, being a function of the following runoff-producing catchment properties:
The runoff curve number method was developed based on daily rainfall P (in.) and its corresponding runoff Q (in.) for the annual floods at a given site. It limits itself to the calculation of runoff depth and does not explicitly account for temporal variations of rainfall intensity. In midsize catchment analysis, the temporal rainfall distribution is introduced at a later stage, during the generation of the flood hydrograph, by means of the convolution of the unit hydrograph with the effective storm hyetograph (Section 5.3).
Runoff Curve Number Equation
In the runoff curve number method:
The method assumes a proportionality between retention and runoff, such that the ratio of actual retention to potential retention is equal to the ratio of actual runoff to potential runoff:
This assumption underscores the conceptual basis of the runoff curve number method, namely the asymptotic behavior of actual retention toward potential retention for sufficiently large values of potential runoff.
For practical applications, Eq. 5-1 is modified by reducing the potential runoff by an amount equal to the initial abstraction Ia. The latter consists mainly of interception, surface storage, and some infiltration, which take place before runoff begins.
Solving for Q from Eq. 5-2:
which is physically subject to the restriction that P > Ia, i.e., the potential runoff minus the initial abstraction cannot be negative.
To simplify Eq. 5-3, the initial abstraction is linearly related to the potential maximum retention as follows:
This relation was obtained based on rainfall-runoff data from small experimental watersheds. The coefficient 0.2 has been subjected to wide scrutiny. For instance, Springer et al.  evaluated small humid and semiarid catchments and found that the coefficient in Eq. 5-4 varied in the range 0.0 to 0.26. Nevertheless, 0.2 is the standard initial abstraction coefficient recommended by NRCS . For research applications and particularly when warranted by field data, it is possible to consider the initial abstraction coefficient as an additional parameter in the runoff curve number method. In general:
in which λ = initial abstraction coefficient.
With Eq. 5-4, Eq. 5-3 reduces to:
which is subject to the restriction that P ≥ 0.2S.
Since potential retention varies within a wide range (0 ≤ S < ∞), it has been conveniently mapped in terms of a runoff curve number, an integer varying in the range 0-100. The chosen mapping equation is:
in which CN is the runoff curve number (dimensionless) and S, 1000 and 10 are given in inches. To illustrate, for CN = 100, S = 0 in.; and for CN = 1, S = 990 in. Therefore, the catchment's capability for rainfall abstraction is inversely proportional to the runoff curve number. For CN = 100, no abstraction is possible, with runoff being equal to total rainfall. On the other hand, for CN = 1 practically all rainfall would be abstracted, with runoff being essentially close to zero.
With Eq. 5-7, Eq. 5-6 can be expressed in terms of CN:
which is subject to the restriction that P ≥ ( 200/ CN ) - 2. In Eq. 5-8, P and Q are given in inches. In SI units, the equation is:
which is subject to the restriction that P ≥ R [ ( 200/CN ) - 2 ]. With R = 2.54 in Eq. 5-9, P and Q are given in centimeters.
For a variable initial abstraction, Eq. 5-8 is expressed as follows:
which is subject to the restriction that P ≥ ( 1000 λ / CN ) - 10 λ. An equivalent equation in SI units is:
A plot of Eqs. 5-8 and 5-9 is shown in Fig. 5-2. This figure is applicable only for the standard initial abstraction value, Ia = 0.2 S. If this condition is relaxed, as in Eqs. 5-10 and 5-11, Fig. 5-2 has to be modified appropriately.
Estimation of Runoff Curve Number From Tables
With rainfall P and runoff curve number CN, the runoff Q can be determined by either Eq. 5-8 or Eq. 5-9, or from Fig. 5-2.
For ungaged watersheds, estimates of runoff curve numbers are given in tables supplied by federal agencies (NRCS, USDA Forest Service) and local city and county departments. Tables of runoff curve numbers for various hydrologic soil-cover complexes are widely available. The hydrologic soil-cover complex describes a specific combination of hydrologic soil group, land use and treatment class, hydrologic surface condition, and antecedent moisture condition. All these have a direct bearing on the amount of runoff produced by a watershed.
The hydrologic soil group describes the type of soil. The land use and treatment class describes the type and condition of vegetative cover. The hydrologic surface condition refers to the ability of the watershed surface to enhance or impede direct runoff. The antecedent moisture condition accounts for the recent history of rainfall and, consequently, it is a measure of the amount of moisture stored by the catchment.
Hydrologic Soil Groups. All soils are classified into four hydrologic soil groups of distinct runoff-producing properties. These groups are labeled A, B, C, and D (Table 5-1).
Group A consists of soils of low runoff potential, having high infiltration rates even when wetted thoroughly. They are primarily deep, very well drained sands and gravels, with a characteristically high rate of water transmission.
Group B consists of soils with moderate infiltration rates when wetted thoroughly, primarily moderately deep to deep, moderately drained to well drained, with moderately fine to moderately coarse textures. These soils have a moderate rate of water transmission.
Group C consists of soils with slow infiltration rate when wetted thoroughly, primarily soils having a layer that impedes downward movement of water or soils of moderately fine to fine texture. These soils have a slow rate of water transmission.
Group D consists of soils of high runoff potential, having very slow infiltration rates when wetted thoroughly. They are primarily clay soils with a high swelling potential, soils with a permanent high water table, soils with a clay layer near the surface, and shallow soils overlying impervious material. These soils have a very slow rate of water transmission.
Maps showing the geographical distribution of hydrologic soil types for most areas in the United States are available either directly from NRCS or from pertinent local agencies.
Additional detail on U.S. soils and their hydrologic soil groups can be found in NRCS publications .
Land Use and Treatment. The effect of the surface condition of a watershed is evaluated by means of land use and treatment classes. Land use pertains to the watershed cover, including every kind of vegetation, litter and mulch, fallow (bare soil), as well as nonagricultural uses such as water surfaces (lakes, swamps, and so on), impervious surfaces (roads, roofs, and the like), and urban areas. Land treatment applies mainly to agricultural land uses, and it includes mechanical practices such as contouring or terracing and management practices such as grazing control and crop rotation. A class of land use/treatment is a combination often found in a catchment.
The runoff curve number method distinguishes between cultivated land, grasslands, and woods and forests. For cultivated lands, it recognizes the following land uses and treatments: fallow, row crop, small grain, close-seed legumes, rotations (from poor to good), straight-row fields, contoured fields, and terraced fields. Additional detail on these land use and treatment classes can be found in reference .
Hydrologic Condition. Grasslands are evaluated by the hydrologic condition of native pasture. The percent of areal coverage by native pasture and the intensity of grazing are visually estimated. A poor hydrologic condition describes less than 50 percent areal coverage and heavy grazing. A fair hydrologic condition describes 50 to 75 percent areal coverage and medium grazing. A good hydrologic condition describes more than 75 percent areal coverage and light grazing.
Woods are small isolated groves or trees being raised for farm or ranch use. The hydrologic condition of woods is visually estimated as follows:
Runoff curve numbers for forest conditions are based on guidelines developed by the U. S. Forest Service. The publication Forest and Range Hydrology Handbook  describes the determination of runoff curve numbers for national and commercial forests in the eastern United States. The publication Handbook of Methods for Hydrologic Analysis  is used for curve number determinations in the forest-range regions in the western United States.
Antecedent Moisture Condition. The runoff curve number method has three levels of antecedent moisture: AMC I, AMC II, and AMC III. The dry antecedent moisture condition (AMC I) has the lowest runoff potential, with the soils being dry enough for satisfactory plowing or cultivation to take place. The average antecedent moisture condition (AMC II) has an average runoff potential. The wet antecedent moisture condition (AMC III) has the highest runoff potential, with the catchment being practically saturated by antecedent rainfalls. Prior to 1993, the appropriate AMC level was based on the total 5-d antecedent rainfall, for dormant or growing season, as shown in Table 5-1. The current version of Chapter 4, NEH-4, released in 1993 , no longer supports Table 5.2, which is included here only for the sake of completeness. Applicable levels of AMC, including fractional values (see Table 5-5), have been developed on a regional basis.
Tables of runoff curve numbers for various hydrologic soil-cover complexes are in current use.
Runoff curve numbers shown in these tables are for the average AMC II condition. Corresponding runoff curve numbers for AMC I and AMC III conditions are shown in Table 5-4.
AMC correlations. Using Eq. 5-7, Hawkins et al  have expressed the values in Table 5-4 in terms of potential maximum retention. They correlated the values of potential maximum retention for AMC I and III with those of AMC II and found the following ratios to be a good approximation:
This led to the following relationships:
which can be used in lieu of Table 5-4 to calculate runoff curve numbers for AMC I and AMC III in terms of the AMC II value.
Estimation of Runoff Curve Numbers from Data
The runoff curve number method was developed primarily for design applications in ungaged catchments and was not intended for simulation of actual recorded hydrographs. In the absence of data, the nationwide tables (Table 5-3) are generally applicable. Where rainfall-runoff records are available, estimations of runoff curve numbers can be obtained directly from data. These values complement and in certain cases may even replace the information obtained from tables.
To estimate runoff curve numbers from data, it is necessary to assemble corresponding rainfall-runoff data sets for several events occurring individually. As far as possible, the events should be of constant intensity and should uniformly cover the entire catchment. The selected set should encompass a wide range of antecedent moisture conditions, from dry to wet. In principle, daily rainfall-runoff data corresponding to the annual floods at a site would result in runoff curve numbers emulating those obtained in the method's original development. Thus, a recommended procedure is to select events that correspond to annual floods. In the absence of a long annual flood series, less selective criteria have been used for candidate storm events, including those of return period less than 1 yr. This choice results in considerable more data for analysis, as well as in curve numbers which are slightly higher than those obtained using an annual flood series. The choice of frequency for candidate storm events is the subject of continuing research.
For each event, a value of P, total rainfall depth, is identified. The associated direct runoff hydrograph is integrated to obtain the direct runoff volume. This runoff volume is divided by the catchment area to obtain Q, the direct runoff depth (in centimeters or inches). The values of P and Q are plotted on Fig. 5-2 and a corresponding value of CN is identified. The procedure is repeated for all events, and a CN value is obtained for each event, as shown in Fig. 5-3. In theory, the AMC II runoff curve number is that which separates the data into two equal groups, with half of the data plotting above the line and half below it. The AMC I runoff curve number is the curve number that envelopes the data from below. The AMC III runoff curve number is the curve number that envelopes the data from above (see Fig. 5-3).
Assessment of Runoff Curve Number Method
The positive features of the runoff curve number method are its simplicity and the fact that runoff curve numbers are related to the major runoff producing properties of the watershed, such as soil type, vegetation type and treatment, surface condition, and antecedent moisture. The method is used in practice to determine runoff depths based on rainfall depths and curve numbers, with no explicit account of rainfall intensity and duration.
A considerable body of experience has been accumulated on the runoff curve number method. Publications continue to appear in the literature either to augment the already extensive experience or to examine critically the applicability of the method to individual situations. For best results, however. the method should be used judiciously, with particular attention paid to its capabilities and limitations.
Experience with the method has shown that results are sensitive to curve number. This stresses the importance of an accurate estimation of curve number to minimize the variance in runoff determinations. The standard tables provide helpful guidelines, but local experience is recommended for increased accuracy. Typical runoff curve numbers used in design are in the range 50 ≤ CN ≤ 98.
Closely associated with the method's sensitivity to runoff curve number is its sensitivity to antecedent moisture. Since runoff curve number varies with antecedent moisture, markedly different results can be obtained for each of the three levels of antecedent moisture. At first, this appears to be a limitation; however, closer examination reveals that runoff is indeed a function of antecedent moisture, with the method's sensitivity to AMC reflecting the conditions likely to prevail in nature. Hjelmfelt et al.  attached a probability meaning to AMC, with AMC I corresponding to 10 percent probability of exceedence, AMC II to 50 percent, and AMC III to 90 percent. This may help explain why practical enveloping curves to determine AMC I and AMC III usually do not encompass all the data.
The popularity of the runoff curve number method is largely due to its simplicity, although proper care is necessary to use the method correctly. The method is essentially a conceptual model to estimate storm runoff volume based on established hydrologic abstraction mechanisms, with the effect of antecedent moisture taken in a probability context. In practice, (average) AMC II describes a typical design condition. When warranted, other antecedent moisture conditions, including those intermediate between I, II, and III, may be considered. An example of regional practice is given in Table 5-5.
Experience with the runoff curve number method has shown that the curve numbers obtained from Table 5-3 tend to be conservative (i.e., too high) for large catchments, especially those located in semiarid and arid regions. Often this is due to the fact that these large catchments have additional sources of hydrologic abstraction, in particular, channel transmission losses, not accounted for by the tables. In this case it is necessary to perform a separate evaluation of the effect of channel abstractions on the quantity of surface runoff.
While the applicability of the runoff curve number procedure appears to be independent of catchment scale, its indiscriminate use for catchments in excess of 250 km2 (100 mi2) without catchment subdivision is generally not recommended. The runoff curve number was originally developed by SCS for use in midsize rural watersheds. Subsequently, the method was applied to small and midsize urban catchments (the TR-55 method). Therefore, its extension to large basins requires considerable judgment.
5.3 UNIT HYDROGRAPH
The concept of unit hydrograph, attributed to Sherman , is used in midsize catchment analysis as a means to develop a hydrograph for a given storm. The word unit is normally taken to refer to a unit depth of effective rainfall or runoff. However, it should be noted that Sherman first used the word to describe a unit depth of runoff (1 cm or 1 in.) lasting a unit increment of time, i.e., an indivisible increment. The unit increment of time can be either 1-h, 3-h, 6-h, 12-h, 24-h, or any other suitable duration. For midsize catchment analysis, unit hydrograph durations from 1 to 6 h are common.
The unit hydrograph is defined as the hydrograph produced by a unit depth of runoff uniformly distributed over the entire catchment and lasting a specified duration. To illustrate the concept of unit hydrograph, assume that a certain storm produces 1 cm of runoff and covers a 50-km2 catchment over a period of 2 h. The hydrograph measured at the catchment outlet would be the 2-h unit hydrograph for this 50-km2 catchment (Fig. 5-4).
A unit hydrograph for a given catchment can be calculated either:
While both methods may be used for gaged catchments, the latter method is appropriate only for ungaged catchments.
Since a unit hydrograph has meaning only in connection with a given storm duration, it follows that a catchment can have several unit hydrographs, each for a different rainfall duration. Once a unit hydrograph for a given duration has been determined, other unit hydrographs can be derived from it by using one of the following methods:
Two assumptions are crucial to the development of the unit hydrograph. These are the principles of linearity and superposition. Given a unit hydrograph, a hydrograph for a runoff depth other than unity can be obtained by simply multiplying the unit hydrograph ordinates by the indicated runoff depth (linearity), as shown in Fig. 5-5 (a). This, of course, is possible only under the assumption that the time base remains constant regardless of runoff depth.
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 lagged in time one increment of unit hydrograph duration, as shown in Fig. 5-5 (b).
The summation of the corresponding ordinates of these hydrographs (superposition) allows the calculation of the composite hydrograph, as shown in Fig. 5-5 (c). The procedure depicted in Fig. 5-5 (a), (b), and (c) is referred to as the convolution of a unit hydrograph with an effective storm hyetograph. In essence, the procedure amounts to stating that the composite hydrograph ordinates are a linear combination of the unit hydrograph ordinates. The composite hydrograph time base is the sum of the unit hydrograph time base minus the unit hydrograph duration plus the storm duration.
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 [1, 4, 15].
The upper limit of applicability of the unit hydrograph is not very well defined. Sherman  used it in connection with basins varying from 1300 to 8000 km2. Linsley et al.  mention an upper limit of 5000 km2 in order to preserve accuracy. More recently, the unit hydrograph has been linked to the concept of midsize catchment, i.e., greater than 2.5 km2 and less than 250 km2. This certainly does not preclude the unit hydrograph technique from being applied to catchments larger than 250 km2, although overall accuracy is likely to decrease with an increase in catchment size.
Development of Unit Hydrographs: Direct Method
To develop a unit hydrograph by the direct method it is necessary to have a gaged catchment, i.e., a catchment equipped with raingages and a stream gage at the outlet, and adequate sets of corresponding rainfall-runoff data; see, for instance, Fig. 5-6.
The rainfall-runoff records should be screened to identify storms suitable for unit hydrograph analysis. Ideally, a storm should have a clearly defined duration, with no rainfall preceding it or following it. The selected storms should be of uniform rainfall intensity both temporally and spatially. In practice, the difficulty in meeting this latter requirement increases with catchment size. As catchment scale grows from midsize to large, the requirement of spatial rainfall uniformity in particular is seldom met. This effectively limits unit hydrograph development by the direct method to midsize catchments.
Catchment Lag. The concept of catchment lag, basin lag, or lag time is central to unit hydrograph analysis. Catchment lag is a measure of the time elapsed between the occurrence of unit rainfall and the occurrence of unit runoff. It is a global measure of response time, encompassing hydraulic length, catchment gradient, drainage density, drainage patterns, and other related factors.
There are several definitions of catchment lag, depending on what particular instant is taken to describe the occurrence of either unit rainfall or runoff. Hall  has identified seven definitions, shown in Fig. 5-7. The T2 lag, defined as the time elapsed from the centroid of effective rainfall to the peak of runoff, is the most commonly used definition of catchment lag.
In unit hydrograph analysis, the concept of catchment lag is used to characterize the catchment response time. Runoff volume must be conserved (i.e., runoff volume should equal one unit of effective rainfall depth). Therefore, short lags result in unit responses featuring high peaks and relatively short time bases; conversely, long lags result in unit responses showing low peaks and long time bases.
In practice, catchment lag is empirically related to catchment characteristics. A general expression for catchment lag is:
in which tl = catchment lag; L = catchment length (length measured along the main stream from outlet to divide); Lc = length to catchment centroid (length measured along the main stream from outlet to a point located closest to the catchment centroid); S = a weighted measure of catchment slope, usually taken as the S2 channel slope (Section 2.3); and C and N are empirical parameters. The parameter L describes length, Lc is a measure of shape, and S relates to relief.
Methodology. In addition to the requirements of uniform rainfall intensity in time and space, storms suitable for unit hydrograph analysis must be of about the same duration. The duration should lie between 10 percent to 30 percent of the catchment lag. The latter requirement implies that runoff response is of the subconcentrated type, with rainfall duration less than time of concentration. Indeed, subconcentrated flow is a characteristic of midsize catchments.
For increased accuracy, direct runoff should be in the range 0.5 to 2.0 units (usually centimeters or inches). Several individual storms (at least five events) should be analyzed to assure consistency. The following steps are applied to each individual storm:
The catchment unit hydrograph is obtained by averaging the unit hydrograph ordinates obtained from each of the individual storms, and averaging the respective unit hydrograph durations. Minor adjustments in hydrograph ordinates may be necessary to ensure that the volume under the unit hydrograph is equal to one unit of runoff depth.
Hydrograph Separation. Only the direct runoff component of the measured hydrograph is used in the computation of the unit hydrograph. Therefore, it is necessary to separate the measured hydrograph into its direct runoff and baseflow components. Interflow, if any, is usually included as part of baseflow.
Procedures for baseflow separation are usually arbitrary in nature. First, it is necessary to identify the point in the receding limb of the measured hydrograph where direct runoff ends. Generally, this ending point is located in such a way that the receding time up to that point is about 2 to 4 times the time-to-peak (Fig. 5-8). For large basins, this multiplier may be greater than 4. As far as possible, the location of the ending point should be such that the time base is an even multiple of the unit hydrograph duration.
A common assumption is that baseflow recedes at the same rate as prior to the storm until the peak discharge has passed, and then it gradually increases to the ending point P in the receding limb, as illustrated by line a in Fig. 5-8. If a stream and groundwater table are hydraulically connected (Fig. 5-9), water infiltrates during the rising limb, reducing baseflow, and exfiltrates during the receding limb, increasing baseflow, as shown by line b in Fig. 5-8 . The most expedient assumption for baseflow separation is a straight line from the start of the rising limb to the ending point, as shown by line c. Differences in baseflow due to the various separation techniques are likely to be small when compared to the direct runoff hydrograph volume.
Other techniques for hydrograph separation and baseflow recession are described in Chapter 11. The development of a unit hydrograph by the direct method is illustrated by Example 5-3.
Development of Unit Hydrographs: Indirect Method
In the absence of rainfall-runoff data, unit hydrographs can be derived by synthetic means. A synthetic unit hydrograph is derived following an established formula, without the need for rainfall-runoff analysis.
For instance, if a triangular shape is assumed, the volume is equal to (Fig. 5-10):
in which V = volume under the triangular unit hydrograph; Qp = peak flow; Tbt = time base of the triangular unit hydrograph; A = catchment area; and (1) = one unit of runoff depth.
From Eq. 5-17:
Synthetic unit hydrograph methods usually relate time base to catchment lag. In turn, catchment lag is related to the timing response characteristics of the catchment, including catchment shape, length, and slope. Therefore, catchment lag is a fundamental variable in synthetic unit hydrograph analysis.
Several methods are available for the calculation of synthetic unit hydrographs.
Two widely used methods, the Snyder and the Natural Resources Conservation Service (NRCS) methods, are described here.
The Clark unit hydrograph, also widely used, is based on catchment routing techniques; therefore, it is described in Chapter 10.
Snyder's Synthetic Unit Hydrograph
In 1938, Snyder  introduced the concept of synthetic unit hydrograph. The analysis of a large number of hydrographs from catchments in the Appalachian region led to the following formula for lag:
in which tl = catchment or basin lag, in hours, L = length along the mainstream from outlet to divide,
Snyder's formula for peak flow is:
which when compared with Eq. 5-18 reveals that
is an empirical coefficient relating triangular time base to lag. Snyder gave values of Cp in the range 0.56 to 0.69, which are associated with Tbt /tl ratios in the range 3.57 to 2.90. The lower the value of Cp (i.e., the lower the peak flow), the greater the value of Tbt /tl and the greater the capability for catchment storage.
In SI units, Snyder's peak flow formula is:
in which Qp = unit hydrograph peak flow corresponding to 1 cm of effective rainfall, in cubic meters per second; A = catchment area, in square kilometers; and tl = lag, in hours. In U.S. customary units, Snyder's peak flow formula is
in which Qp = unit hydrograph peak flow corresponding to 1 in. of effective rainfall in cubic feet per second;
In Snyder's method, the unit hydrograph duration is a linear function of the lag:
in which tr = unit hydrograph duration.
In applying the procedure to flood forecasting, Snyder recognized that the actual duration of the storm is usually greater than the duration calculated by Eq. 5-24. Therefore, he devised a formula to increase the lag in order to account for the increased storm duration. This led to:
in which tlR is the adjusted lag corresponding to a duration tR.
Assuming uniform effective rainfall for simplicity, the unit hydrograph time-to-peak is equal to one-half of the storm duration plus the lag (Fig. 5-7). Therefore, the time-to-peak in terms of the lag is:
When calculating the actual time base of the unit hydrograph, Snyder included interflow as part of direct runoff. This results in a longer time base than that corresponding only to direct runoff. Snyder's formula for actual time base is the following:
in which Tb = actual unit hydrograph time base (including interflow), in hours
and tl = lag, in hours.
For a 24-h lag, this formula gives Tb /tl = 6, which is a reasonable value considering that interflow is being included in the calculation.
For smaller lags, however, Eq. 5-27 gives unrealistically high values of
The Snyder method gives peak flow (Eq. 5-22), time-to-peak (Eq. 5-26), and time base (Eq. 5-27) of the unit hydrograph.
These values can be used to sketch the unit hydrograph, adhering to the requirement that unit hydrograph volume should equal 1 unit of runoff depth.
Snyder gave a distribution chart to aid in plotting the unit hydrograph ordinates, but cautioned against the exclusive reliance on this graph to develop the shape of the unit hydrograph (Fig. 5-11).
The Snyder method has been extensively used by the U.S. Army Corps of Engineers. Their experience has led to two empirical formulas that aid in determining the shape of the Snyder unit hydrograph :
in which W50 = width of unit hydrograph at 50 percent of peak discharge in hours; W75 = width of unit hydrograph at 75 percent of peak discharge in hours; Qp = peak discharge in cubic meters per second; and A = catchment area in square kilometers (Fig. 5-12). These time widths should be proportioned in such a way that one-third is located before the peak and two-thirds after the peak.
Snyder cautioned that lag may tend to vary slightly with flood magnitude and that synthetic unit hydrograph calculations are likely to be more accurate for fan-shaped catchments than for those of highly irregular shape. He recommended that the coefficients Ct and Cp be determined on a regional basis.
The examination of Eq. 5-19 reveals that Ct is largely a function of catchment slope, since both length and shape have already been accounted for in L and Lc , respectively. Since Eq. 5-19 was derived empirically, the actual value of Ct depends on the units of L and Lc. Furthermore, Eq. 5-19 implies that when the product LLc is equal to 1, the lag is equal to Ct. Since for two catchments of the same size, lag is a function of slope, it is unlikely that Ct is a constant. To give an example, an analysis of 20 catchments in the north and middle Atlantic United States led to : Ct = 0.6/S1/2. A similar conclusion is drawn from Eq. 5-16. Therefore, values of Ct have regional meaning, in general being a function of catchment slope. Values of Ct quoted in the literature reflect the natural variability of catchment slopes.
The parameter Cp is dimensionless and varies within a narrow range. In fact, it is readily shown that the maximum possible value of Cp is 11/12. Since triangular time base cannot be less than twice the time-to-peak (otherwise, runoff diffusion would be negative, clearly a physical impossibility), it follows that in the limit (i.e., in the absence of runoff diffusion), Tbt = 2tp; and, therefore, Cp = tl / tp = 11/12. In practice, triangular time base is usually about 3 times the time-to-peak. For Tbt = 3tp, a similar calculation leads to: Cp = 0.61, which lies approximately in the middle of Snyder's data (0.56-0.69).
Since Ct increases with catchment storage and Cp decreases with catchment storage, the ratio
NRCS Synthetic Unit Hydrograph
The NRCS synthetic unit hydrograph is the dimensionless unit hydrograph developed by Victor Mockus in the 1950s . This hydrograph was developed based on the analysis of a large number of natural unit hydrographs from a wide range of catchment sizes and geographic locations. The method has come to be recognized as the NRCS synthetic unit hydrograph and has been applied to midsize catchments throughout the world.
The method differs from Snyder's in that it uses a constant ratio of triangular time base to time-to-peak,
To calculate catchment lag (the T2 lag), the NRCS method uses the following two methods:
The curve number method is limited to catchments of areas less than 8 km2 (2000 ac), although recent evidence suggests that it may be extended to catchments up to 16 km2 (4000 ac) . In the curve number method, the lag is expressed by the following formula:
in which tl = catchment lag, in hours; L = hydraulic length (length measured along principal watercourse), in meters; CN = runoff curve number; and Y = average catchment land slope, in meters per meter. In U.S. customary units, the formula is:
The velocity method is used for catchments larger than 8 km2,
or for curve numbers outside of the range
in which tl = lag, and tc = time of concentration. NRCS experience has shown that this ratio is typical of midsize catchments .
In the NRCS method, the ratio of time-to-peak to unit hydrograph duration is fixed at
which is close to Snyder's ratio of 6. Assuming uniform effective rainfall for simplicity, the time-to-peak is, by definition, equal to
Eliminating tr from Eqs. 5-33 and 5-34, leads to
To derive the NRCS unit hydrograph peak flow formula, the ratio Tbt /tp = 8/3 is used in Eq. 5-18, leading to
In SI units, the peak flow formula is:
in which Qp = unit hydrograph peak flow for 1 cm of effective rainfall in cubic meters per second; A = catchment area, in square kilometers; and tp = time-to-peak, in hours. In U.S. customary units, the NRCS peak flow formula is:
in which Qp = unit hydrograph peak flow for 1 in. of effective rainfall; A = catchment area, in square miles; and tp = time-to-peak, in hours.
Given Eqs. 5-32 and 5-34, the time-to-peak can be readily calculated
as follows: tp = 0.5tr + 0.6tc. Once tp and Qp have been determined, the NRCS dimensionless unit hydrograph (Fig. 5-13) is used to calculate the unit hydrograph ordinates.
The shape of the dimensionless unit hydrograph is more in agreement with unit hydrographs that are likely to occur in nature than the triangular shape
(Tbt / tp = 8/3) used to develop the
peak flow value.
The dimensionless unit hydrograph has a value of Tb / tp = 5.
Values of NRCS dimensionless unit hydrograph ordinates at intervals of 0.2 (t /tp) are given in Table 5-7.
The calculation of an NRCS synthetic unit hydrograph is illustrated by the following example.
Two-parameter NRCS method. The NRCS method provides a unit hydrograph shape and, therefore, leads to more reproducible results than the Snyder method. However, the ratio Tbt /tp is kept constant and equal to 8/3. Also, when lag is calculated by the velocity method, the ratio tl /tc is kept constant and equal to 6/10. Although these assumptions are based on a wide range of data, they render the method inflexible in certain cases.
In particular, values of Tbt /tp other than 8/3 may lead to other shapes of unit hydrographs. Larger values of Tbt /tp (equivalent to lower values of Cp in the Snyder method) imply greater catchment storage. Therefore, since the NRCS method fixes the value of Tbt /tp, it should be limited to midsize catchments in the lower end of the range (2.5 - 250 km2). The Snyder method, however, by providing a variable Tbt /tp may be used for larger catchments .
Efforts to extend the range of applicability of the NRCS method
have led to the relaxation of the
which converts the NRCS method into a two-parameter model like the Snyder method, thereby increasing its flexibility.
Other Synthetic Unit Hydrographs
The Snyder and NRCS methods base their calculations on the following properties:
In addition, the NRCS method specifies a gamma function for the shape of the unit hydrograph. Many other synthetic unit hydrographs have been reported in the literature . In general, any procedure defining geometric properties and hydrograph shape can be used to develop a synthetic unit hydrograph.
Change in Unit Hydrograph Duration
A unit hydrograph, whether derived by direct or indirect means, is valid only for a given (effective) storm duration. In certain cases, it may be necessary to change the duration of a unit hydrograph. For instance, if an X-hour unit hydrograph is going to be used with a storm hyetograph defined at Y-hour intervals, it is necessary to convert the X-hour unit hydrograph into a Y-hour unit hydrograph.
In general, once a unit hydrograph of a given duration has been derived for a catchment, a unit hydrograph of another duration can be calculated. There are two methods to change the duration of unit hydrographs:
The superposition method converts an X-hour unit hydrograph into a nX-hour unit hydrograph, in which n is an integer. The S-hydrograph method converts an X-hour unit hydrograph into a Y-hour unit hydrograph, regardless of the ratio between X and Y.
Superposition Method. This method allows the conversion of an X-hour unit hydrograph into a nX-hour unit hydrograph, in which n is an integer. The procedure consists of lagging nX-hour unit hydrographs in time, each for an interval equal to X hours, summing the ordinates of all n hydrographs, and dividing the summed ordinates by n to obtain the nX-hour unit hydrograph. The volume under X-hour and nX-hour unit hydrographs is the same. If Tb is the time base of the X-hour hydrograph, the time base of the nX-hour hydrograph is equal to Tb + (n - 1)X. The procedure is illustrated by the following example.
S-Hydrograph Method. The S-hydrograph method allows the conversion of an X-hour unit hydrograph into a Y-hour unit hydrograph, regardless of the ratio between X and Y. The procedure consists of the following steps:
The volume under X-hour and Y-hour unit hydrographs is the same. If Tb is the time base of the X-hour unit hydrograph, the time base of the Y-hour unit hydrograph is Tb - X + Y. The procedure is illustrated by the following example.
Minor errors in unit hydrograph ordinates may often lead to errors (i.e., undesirable oscillations) in the resulting S-hydrograph. In this case, a certain amount of smoothing may be required to achieve the typical S-shape (Fig. 5-14).
Convolution and Composite Hydrographs
The procedure to derive a composite or flood hydrograph based on a unit hydrograph and an effective storm hyetograph is referred to as hydrograph convolution. This technique is based on the principles of linearity and superposition. The volume under the composite hydrograph is equal to the total volume of the effective rainfall. If Tb is the time base of the X-hour unit hydrograph and the storm consists of n X-hour intervals, the time base of the composite hydrograph is equal to Tb - X + nX = Tb + (n - 1)X. The convolution procedure is illustrated by the following example.
Unit Hydrographs from Complex Storms
The convolution procedure enables the calculation of a storm hydrograph based on a unit hydrograph and a storm hyetograph. In theory, the procedure can be reversed to allow the calculation of a unit hydrograph for a given storm hydrograph and storm hyetograph.
Method of Forward Substitution. The unit hydrograph can be calculated directly due to the banded property of the convolution matrix (see Table 5-11). With m = number of nonzero unit hydrograph ordinates, n = number of intervals of effective rainfall, and N = number of nonzero storm hydrograph ordinates, the following relation holds:
By elimination and back substitution, the following formula can be developed for the unit hydrograph ordinates ui as a function of storm hydrograph ordinates qi and effective rainfall depths rk :
for i varying from 1 to m. In the summation term, j decreases from j -1 to 1, and k increases from 2 up to a maximum of n.
This recursive equation allows the direct calculation of a unit hydrograph based on hydrographs from complex storms. In practice, however, it is not always feasible to arrive at a solution because it may be difficult to get a perfect match of storm hydrograph and effective rainfall hyetograph (due to errors in the data). For one thing, the measured storm hydrograph would have to be separated into direct runoff and baseflow before attempting to use Eq. 5-44.
The uncertainties involved have led to the use of the least square technique. In this technique, rainfall-runoff data (r,h) for a number of events are used to develop a set of average values of u using statistical tools . Other methods to derive unit hydrographs for complex storms are discussed by Singh .
5.4 TR-55 METHOD
The TR-55 method is a collection of simplified procedures developed by the USDA Natural Resources Conservation Service to calculate peak discharges, storm hydrographs, and stormwater storage volumes in small/midsize urban catchments . It consists of three methodologies:
The graphical method calculates a flood peak discharge for a hydrologically homogeneous catchment, i.e., that which can be represented by a single area, of given slope and curve number. The tabular method calculates a flood hydrograph for a hydrologically heterogeneous catchment, which is better analyzed by dividing it into several homogeneous subareas, each of given slope and curve number. These methods were developed based on information obtained with the NRCS TR-20 hydrologic computer model (Section 13.4). They are designed to be used in cases where their applicability can be clearly demonstrated, in lieu of more elaborate techniques. Whereas TR-55 does not specify catchment size, the graphical method is limited to catchments with time of concentration in the range 0.1-10 h. This encompasses most small and midsize catchments in the terminology used in this book. Likewise, the tabular method is limited to catchments with time of concentration in the range 0.1-2 h.
The graphical method is described in this section. The tabular method is described in the original reference . The detention-basin method is described in Section 8.5.
TR-55 Storm, Catchment and Runoff Parameters
Rainfall in TR-55 is described in terms of total rainfall depth and one of four standard 24-h temporal rainfall distributions: Type I, Type IA, Type II, and Type III. These distributions are shown in Fig. 5-15. Type I applies to California (south of the San Francisco Bay area) and Alaska; Type IA applies to the Pacific Northwest and Northern California; Type III applies to the Gulf Coast states; and Type II applies everywhere else within the contiguous United States, as shown in Fig. 5-16.
The duration of these rainfall distributions is 24 h.
This constant duration was selected because most rainfall data is reported on a 24-h basis.
Rainfall intensities corresponding to durations shorter than
TR-55 uses the runoff curve number method (Section 5.1) to abstract total rainfall depth and calculate runoff depth. The abstraction procedure follows established guidelines , with extensions to account for curve numbers applicable to urban areas. In addition, TR-55 includes procedures to determine the time of concentration for the following types of surface flow:
Shallow concentrated flow is a type of flow of characteristics in between those of overland flow and streamflow.
Applicability of TR-55
When using TR-55, there is a choice between graphical or tabular method. The graphical method gives only a peak discharge, whereas the tabular method provides a flood hydrograph. The graphical method should be used for hydrologically homogeneous catchments; the tabular method should be used for hydrologically heterogeneous catchments, for which catchment subdivision is necessary.
The primary objective of TR-55 is to provide simplified techniques, thereby reducing the effort involved in routine hydrologic calculations. The potential accuracy of the method is less than that which could be obtained with more elaborate techniques. The method is strictly applicable to surface flow and should not be used to describe flow properties in underground conduits.
Selection of Runoff Curve Number
To estimate curve numbers for urban catchments, TR-55 defines two types of areas:
Once pervious and impervious areas are delineated, the percent imperviousness can be determined. Impervious areas are of two kinds:
The question is: Do the impervious areas connect directly to the drainage system, or do they discharge onto lawns or other pervious areas where infiltration can occur?
An impervious area is considered connected:
An impervious area is considered unconnected if runoff from it spreads over a pervious area as overland (sheet) flow.
Table 5-3 (a) shows urban runoff curve numbers for connected impervious areas. The curve numbers shown are for typical values of average percent impervious area (second column). These composite curve numbers were developed based on the following assumptions:
Tables 5-3 (b), (c), and (d) show runoff curve numbers for cultivated agricultural lands, other agricultural lands, and arid and semiarid rangelands, respectively.
Figure 5-17 is used in lieu of Table 5-3 (a) when the average percent (connected) impervious area and/or pervious area land use assumptions are other than those shown in the table. For example, Table 5-3 (a) gives a CN = 70 for a 1/2-acre lot in hydrologic soil group B, assuming a 25 percent impervious area. If the lot has a different percent impervious area, say 20 percent, but the pervious area land use is the same as that assumed in Table 5-3 (a) (open space in good hydrologic condition), then the pervious area CN is 61 (for hydrologic soil group B) and the composite curve number obtained from Fig. 5-17 with 20 percent impervious area and pervious area CN = 61 is: CN = 69. The difference between 70 and 69 reflects the difference in percent impervious area only (25 vs 20 percent).
Figure 5-18 is used to determine a composite CN when all or part of the impervious area is unconnected and the percent imperviousness is 30 percent or less. However, when the percent imperviousness is more than 30 percent, Fig. 5-17 is used instead to determine the composite CN, since the absorptive capacity of the remaining pervious areas (less than 70 percent) will not significantly affect runoff. In Fig. 5-18, enter the right-side figure with percent imperviousness to the line matching the ratio of unconnected impervious to total impervious area. Then, move horizontally to the left-side figure to match the pervious area CN, and vertically down to find the composite CN. For example, for a 1/2-acre lot with 20 percent imperviousness, 75 percent of which is unconnected, and pervious CN = 61, the composite CN (from Fig. 5-18) is: CN = 66. If all of the impervious area is connected (i.e., zero percent unconnected), the resulting CN (from Fig. 5-17) is: CN = 69. This value matches the example of the previous paragraph.
Travel Time and Time of Concentration
For any reach or subreach, travel time is the ratio of flow length to flow velocity. The time of concentration is the sum of travel times through the individual subreaches.
For overland (sheet) flow with length less than 300 ft, TR-55 uses the following formula for travel time:
in which tt, = travel time, in hours; n = Manning n; L = flow length, in feet; P2 = 2-y 24-h rainfall depth in inches; and S = average land slope, in feet per foot. In SI units, this equation is:
in which L is given in meters; P2 in centimeters; S in meters per meter; and the remaining terms are the same as in Eq. 5-45. TR-55 values of Manning n applicable to overland flow are given in Table 5-12.
Overland flow lengths over 300 ft (90 m) lead to a form of surface flow referred to as shallow concentrated flow. In this case, the average flow velocity is determined from Fig. 5-19. For streamflow, the Manning equation (Eq. 2-65) can be used to calculate average flow velocities. Values of Manning n applicable to open channel flow are obtained from standard references [2, 3, 6].
TR-55 Graphical Method
The TR-55 graphical method calculates peak discharge based on the concept of unit peak flow. The unit peak flow is the peak flow per unit area, per unit runoff depth. In TR-55, unit peak flow is a function of the following variables:
Peak discharge is calculated by the following formula:
in which Qp = peak discharge in L3T-1 units; qu = unit peak flow in T-1 units; A = catchment area in L2 units; Q = runoff depth in L units; and F = surface storage correction factor (dimensionless).
To use the graphical method, it is first necessary to evaluate the catchment flow type and to calculate the time of concentration, assuming either: (1) overland flow, (2) shallow concentrated flow, or (3) streamflow. The runoff curve number is determined from either Table 5-3, Fig. 5-17, or Fig. 5-18. A flood frequency is selected, and an appropriate rainfall map (depth-duration-frequency) is used to determine the rainfall depth for the 24-h duration and the chosen frequency. With the rainfall depth P and the CN, the runoff depth Q is determined using either Fig. 5-2, Eq. 5-8, or Eq. 5-9.
The initial abstraction is calculated by combining Eqs. 5-4 and 5-7 to yield:
in which Ia = initial abstraction, in inches. The equivalent SI formula is:
in which Ia is given in centimeters.
The surface storage correction factor F is obtained from Table 5-13 as a function of the percentage of pond and swamp areas.
With time of concentration tc, ratio Ia/P, and storm type (either I, IA, II, or III), Fig. 5-20 is used to determine the unit peak flow in cubic feet per second per square mile per inch. Interpolation can be used for values of Ia/P different than those shown in Fig. 5-20. For values of Ia/P outside of the range shown in Fig. 5-20, the maximum (or minimum) value should be used.
Peak discharge is calculated by Eq. 5-47 as a function of unit peak flow, catchment area, runoff depth, and surface storage correction factor. The TR-55 graphical method is limited to runoff curve numbers greater than 40, with time of concentration in the range 0.1 to 10 h, and surface storage areas spread throughout the catchment and covering less than 5 percent of it. The computational procedure is illustrated by the following examples.
Assessment of TR-55 Graphical Method
The TR-55 graphical method provides peak discharge as a function of unit peak flow, catchment area, runoff depth, and surface storage correction factor. The unit peak flow is a function of time of concentration, abstraction parameter Ia/P, and NRCS storm type. The runoff depth is a function of total rainfall depth and runoff curve number.
In the TR-55 graphical method, time of concentration accounts for both runoff concentration and runoff diffusion. From Fig. 5-20, it is clearly seen that unit peak flow decreases with time of concentration, implying that the longer the time of concentration, the greater the catchment storage and peak flow attenuation.
The parameter Ia/P is related to the catchment's abstractive properties. The greater the curve number, the lesser the value of Ia/P and the greater the unit peak flow. The surface storage correction factor F reduces the peak discharge to account for additional runoff diffusion caused by surface storage features typical of low relief catchments (i.e., ponds and swamps). The geographical location and associated storm type is accounted for by the four standard NRCS temporal storm distributions. Therefore, the TR-55 graphical method accounts for hydrologic abstraction, runoff concentration and diffusion, geographical location and type of storm, and the additional surface storage of low-relief catchments.
The TR-55 graphical method may be considered an extension of the rational method to midsize catchments. The unit peak flow used in the graphical method is similar in concept to the runoff coefficient of the rational method. However, unlike the latter, the TR-55 graphical method includes runoff curve number and storm type and is applicable to midsize catchments with times of concentration up to 10 h.
The unit values of catchment area, runoff depth, and time of concentration may be used to provide a comparison between the TR-55 graphical method and the rational method. To illustrate, assume a catchment area of 1 mi2 (640 ac), time of concentration 1 h, corresponding rainfall intensity 1 in./h, and runoff coefficient C = 0.95 (the maximum practicable value). A calculation by Eq. 4-4 gives a peak discharge of Qp = 613 ft3/s.
A calculation with the TR-55 graphical method, using the lowest possible value of abstraction for comparison purposes (Ia/P = 0.10), gives the following results: For storm type I, 203 ft3/s; type IA, 108 ft3/s; type II, 360 ft3/s; and type III, 295 ft3/s. This example shows the effect of regional storm hyetograph on the calculated peak discharge. It also shows that the TR-55 graphical method generally gives lower peak flows than the rational method. This may be attributed to the fact that the TR-55 method accounts for runoff diffusion in a somewhat better way than the rational method. However, it should be noted that the peak discharges calculated by the two methods are not strictly comparable, since the value of Ia/P = 0.1 does not correspond exactly to C = 0.95.
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