5.2  RUNOFF CURVE NUMBER METHOD

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5.3  UNIT HYDROGRAPH TECHNIQUES

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The concept of unit hydrograph, originated by Sherman [14], is used in midsize catchment analysis as a means to develop a hydrograph for any 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 catchments, 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).

Figure 5-4  Concept of a unit hydrograph.

A unit hydrograph for a given catchment can be calculated either

1. Directly, by using rainfall-runoff data for selected events, or

2. indirectly, by using a synthetic unit hydrograph formula.

While both methods can be used for gaged catchments, only the latter method is appropriate 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:

1. Superposition method and

2. S-hydrograph method.

Two assumptions are crucial to the development of the unit hydrograph technique. 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). This, of course, is possible only under the assumption that the time base remains constant regardless of runoff depth (Fig. 5-5(a)).

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 (Fig. 5-5(b)). The summation of the corresponding ordinates of these hydrographs (superposition) allows the calculation of the composite hydrograph (Fig. 5-5(c)). The procedure depicted in Fig. 5-5 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, while 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

(a) (b) (c) Figure 5-5  Unit hydrograph properties: (a) linearity; (b) lagging; (c) superposition.

functions of flow depth and stage. In practice, the linear assumption provides a convenient means of calculating runoff response without the complexities associated with nonlinear analysis. More recent developments of unit hydrograph theory [1, 4, 15] have relaxed the linear assumption. Methods accounting for the nonlinearity of runoff response constitute what is known as nonlinear unit hydrograph theory.

The upper limit of applicability of the unit hydrograph is not very well defined. Sherman [14] used it in connection with basins varying from 1300 to 8000 km². Linsley et al. [10] 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 cenainly 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 area.

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 rain gages and a stream gage at the outlet, and adequate sets of corresponding rainfall-runoff data.

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 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 the development of unit hydrograph theory. It is a measure of the time elapsed between the occurrence of unit rainfall and the occurrence of unit runoff. Catchment lag 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 [7] has identified seven definitions, shown in Fig. 5-6. Among them 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 respone time. Because runoff volume must be conserved (i.e., runoff volume should equal one unit of effective rainfall depth), short lags result in unit responses featuring hig 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 the following:

L Lc   N tl = C (______)              S1/2

(5-16)

in which t_t = catchment lag; L = catchment length (length measured along the main

Figure 5-6  Alternate definitions of catchment lag.

stream from outlet to divide); L_c = 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 (Chapter 2); and C and N are empirical parameters. The parameter L describes length, L_c 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 should be of about the same duration. The duration should lie between 10 percent to 30 percent of catchment lag. The latter requirement implies that runoff response is of the sub concentrated type, with rainfall duration less than time of concentration. Indeed, sub concentrated flow is a characteristic of midsize catchments.

For increased accuracy, direct runoff should be in the range O.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:

1. Separation of the measured hydrograph into direct runoff hydrograph (DRR) and baseflow (BF), following the procedures explained below.

2. Calculation of direct runoff volume (DRV) by integrating the direct runoff hydrograph (DRH).

3. Calculation of direct runoff depth (DRD) by dividing the direct runoff volume (DRV) by the catchment area.

4. Calculation of unit hydrograph (UR) ordinates by dividing the ordinates of the direct runoff hydrograph (DRR) by the direct runoff depth (DRD).

5. Estimation of the unit hydrograph duration.

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 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. 5-7). 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 gradually increases to the ending point P in the receding limb, as illustrated by line a in Fig. 5-7. If a stream and groundwater table are hydraulically connected (Fig. 5-8), water infiltrates during the rising limb, reducing baseflow, and exfiltrates during the receding limb, increasing

Figure 5-7  Procedures for baseflow separation.

Figure 5-8  Hydraulically connected stream and water table.

baseflow, as shown by line b in Fig. 5-7 [5]. 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. Additional methods for hydrograph separation and baseflow recession are described in Chapter 11.

Example 5-2. A unit hydrograph is to be developed for a 37.8-km2 catchment with a lag time of 12 h. A 2-h rainfall produced the following streamflow data: Time (h) 0 2 4 6 8 10 12 14 16 18 20 22 24 Streamflow (m3/s) 2 1 3 5 9 8 7 6 5 4 3 1 1 Develop a unit hydrograph for this catchment. A summary of the calculations is shown in Table 5-5. Columns 1 and 2 show time and measured streamflow, respectively. Baseflow is established by examining the measured streamflow. Since the hydrograph rise starts at 2 h and ends at 22 h, a value of baseflow equal to 1 m3/s appears reasonable. (In practice, a more detailed analysis as described in Section 11.5 may be necessary.) Column 3 shows the ordinates of the DRH obtained by substracting baseflow from the measured streamflow. To calculate direct runoff depth, Table 5-5 Development of Unit Hydrograph: Direct Method, Example 5-2. (1) (2) (3) (4) (5) (6) (7) Time (h) Streamflow (m3/s) DRH (m3/s) Simpson's coefficients Volume UH (m3/s) Verification 0 2 __ __ __ __ __ 2 1 0 1 0 0.00 0.00 4 3 2 4 8 2.50 10.00 6 5 4 2 8 5.00 10.00 8 9 8 4 32 10.00 40.00 10 8 7 2 14 8.75 17.50 12 7 6 4 24 7.50 30.00 14 6 5 2 10 6.25 12.50 16 5 4 4 16 5.00 20.00 18 4 3 2 6 3.75 7.50 20 3 2 4 8 2.50 10.00 22 1 0 1 0 0.00 0.00 24 1 __ __ __ __ __ Sum 126 157.50 DRH is integrated numerically following Simpson's rule. The Simpson's rule coefficients are shown in Col. 4. Column 5 shows the weighted ordinates obtained by multiplying Col. 3 by Col. 4. Summing up the weighted ordinates (Col. 5); a value of 126 m3/s is obtained. Since the integration interval is 2 h, the DRV (according to Simpson's rule) is DRV = (126 m3/s X 7200 seconds)/ 3 = 302,400 m3. The DRD is obtained by dividing DRV by the catchment area (37.8 km2) to yield: DRD = 0.8 cm. The unit hydrographs ordinates (Col. 6) are calculated by dividing the DRH ordinates (Col. 3) by DRD. To verify the calculations, the unit hydrograph shown in Col. 6 is integrated by multiplying Col. 4 times Col. 6 to obtain Col. 7. The sum of Col. 7 is 157.5 m3/s. It is verified that the ratio of DRV to unit hydrograph volume is indeed 0.8 (126 / 157.5 = 0.8). Finally, it is confirmed that the unit hydrograph duration (2 h) is an appropriate percentage (17 percent) of the lag time (12 h).

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 a unit hydro graph derived following an established formula, without the need for rainfall-runoff data analysis.

The development of synthetic unit hydrographs is based on the following principle: Since the volume under the hydrograph is known (volume is equal to catchment area multiplied by 1 unit of runoff depth), the peak discharge can be calculated by assuming a certain unit hydrograph shape. For instance, if a triangular shape is assumed (Fig. 5-9), the volume is equal to:

QpTbt V = _______ = A x (1)           2

(5-17)

in which V = volume under the triangular unit hydrograph; Q_p = peak flow; T_bt = time base of the triangular unit hydrograph; A = catchment area; and (1) = one unit of runoff depth. From Eq. 5-17:

Figure 5-9  Triangular unit hydrograph.

2A Qp = _____           Tbt

(5-18)

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 parameter in synthetic unit hydrograph determinations.

Several methods are available for the calculation of synthetic unit hydrographs. Two widely used methods, the Snyder and the Soil Conservation Service methods, are described here. The Clark method, also widely used, is based on catchment routing techniques; therefore, it is described in Chapter 10.

Snyder's Synthetic Unit Hydrograph

In 1938, Snyder [17] 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:

tl = Ct (L Lc)0.3

(5-19)

in which t_l = catchment or basin lag in hours, L = length along the mainstream from outlet to divide, L_c = length along the mainstream from outlet to point closest to catchment centroid, and C_t = a coefficient accounting for catchment gradient and associated catchment storage. With distances L and L_c in kilometers, Snyder gave values of C_t varying in the range 1.35 to 1.65, with a mean of 1.5. With distances L and L_c in miles, the corresponding range of C_t is 1.8 to 2.2, with a mean of 2.

Snyder's formula for peak flow is:

Cp A Qp = _______             tl

(5-20)

which when compared with Eq. 5-18 reveals that

2 Cp = ______           Tbt          _____            tl

(5-21)

is an empirical coefficient relating triangular time base to lag. Snyder gave values of C_p in the range 0.56 to 0.69, which are associated with T_bt / t_l ratios in the range 3.57 to 2.90. The lower the value of C_p (i.e., the lower the peak flow), the greater the value of T_bt / t_l and the greater the capability for catchment storage.

In SI units, Snyder's peak flow formula is:

2.78Cp A Qp = __________               tl

(5-22)

in which Q_p = unit hydrograph peak flow corresponding to 1 cm of effective rainfall, in cubic meters per second; A = catchment area, in square kilometers; and t_l = lag, in hours. In U.S. customary units, Snyder's peak flow formula is

645Cp A Qp = __________               tl

(5-23)

in which Q_p = unit hydrograph peak flow corresponding to 1 in. of effective rainfall in cubic feet per second; A = catchment area in square miles; and t_l = lag in hours.

In Snyder's method, the unit hydrograph duration is a linear function of lag:

tr = (2/11) tl

(5-24)

in which t_r = 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:

tR - tr tlR = tl + ________                    4

(5-25)

in which t_lR is the adjusted lag corresponding to a duration t_R.

Assuming uniform effective rainfall for simplicity, the unit hydrograph time-topeak 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:

tp = (12/11) tl

(5-26)

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:

Tb = 72 + 3tl

(5-27)

in which T_b = actual unit hydrograph time base (including interflow) in hours and t_l = lag in hours. For a 24-h lag, this formula gives T_b / t_l = 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 T_b / t_l. For instance, for a 6-h lag, T_b / t_l = 15. For midsize catchments, and excluding interflow, experience has shown that values of T_b / t_p around 5 (corresponding to values of T_b / t_l, around 5.45) may be more realistic.

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 (Fig. 5-10) 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.

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 [20]:

6.33 W50 = ____________             (Qp / A)1.08

(5-28)

Figure 5-10  Snyder's distribution chart for plotting unit hydrograph orinates [17].

3.58 W75 = ____________             (Qp / A)1.08

(5-29)

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; Q_p = peak discharge in cubic meters per second; and A = catchment area in square kilometers (Fig. 5-11). 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 fanshaped catchments than for those of highly irregular shape. He recommended that the coefficients C_t, and C_p be determined on a regional basis.

Examination of Eq. 5-19 reveals that C, is largely a function of catchment slope,

Figure 5-11  Snyder's synthetic unit hydrograph widths: W50 and W75 .

since both length and shape have already been accounted for in L and L_c , respectively. Since Eq. 5-19 was derived empirically, the actual value of Cr depends on the units of L and L_c. Furthermore, Eq. 5-19 implies that when the product of (LL_c is equal to 1, the lag is equal to C_t. Since for two catchments of the same size, lag is a function of slope, it is unlikely that C_t is a constant. To give an example, an analysis of 20 catchments in thep.orth-and middle Atlantic United States [19] led to: C_t = 0.6/S^1/2. A similar conclusion is drawn from Eq. 5-16. Therefore, values of C_t have regional meaning, in general being a function of catchment slope. Values of C_t quoted in the literature reflect the natural variability of catchment slopes.

The parameter C_p is dimensionless and varies within a narrow range. In fact, it is readily shown that the maximum possible value of C_p 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 impossipility), it follows that in the limit (Le., absence of runoff diffusion), T_bt = 2t_p; and, therefore, C_p = t_l / t_p = 11/12. In practice, triangular time base is usually about 3 times the time-to-peak. For T_bt = 3t_p , a similar calculation leads to: C_p = 0.61, which lies approximately' in the middle of Snyder's data (0.56-0.69).

Since C_t increases with catchment storage and C_p decreases with catchment storage, the ratio C_t / C_p can be directly related to catchment storage. Furthermore, the reciprocal ratio (C_p / C_t) can be directly related to extent of urban development, since the latter usually results in a substantial reduction in the catchment's storage capability [26].

Example 5-3. Calculate the properties of a Snyder unit hydrograph using the following data: L = 25 km, Lc = 10 km, A = 400 km2, Ct = 1.5, and Cp = 0.61. Using Eq. 5-19, tt = 7.86 h. From Eq. 5-21, solving for Tbt , Tbt = 25.77 h. Using Eq. 5-22, Qp = 86.3 mJ/s. Using Eq. 5-24, tr = 1.43 h. Using Eq. 5-26, tp = 8.57 h. The time base calculated by Eq. 5-27 is Tb = 95.58 h. This is too high a value. Instead, assume time Tb = 5tp; then: Tb = 42.85 h. Using Eq. 5-28, W50 = 33.2 h; using Eq. 5-29, W75 = 18.8 h . The actual unit hydrograph is drawn primarily on the basis of Qp,tp and Tb with the remaining values used as guidelines.

NRCS Synthetic Unit Hydrograph

The NRCS Synthetic unit hydrograph is the dimensionless unit hydrograph developed by Victor Mockus in the 1950s and described in NEH-4 [21]. 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, T_bt / t_p = 8/3 which implies that C_p is constant and equal to 0.6875. Unlike Snyder's method, the NRCS Meththod uses a constant ratio of actual time base to time-to-peak, T_b / t_p = 5. In addition, it uses a dimensionless hydrograph function to provide a standard unit hydrograph shape.

To calculate catchment lag (the T2 lag), the NRCS Method uses the following two methods:

1. The curve number method and

2. The velocity method.

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) [11]. In the curve number method, the lag is expressed by the following formula:

L0.8( 2540 - 22.86CN )0.7 tl = __________________________             14,104CN 0.7Y 0.5

(5-30)

in which t_l = 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:

L0.8( 1000 - 9CN )0.7 tl = ________________________             1900CN 0.7Y 0.5

(5-31)

The velocity method is used for catchments larger than 8 km², or for curve numbers outside of the range 50 to 95. The main stream is divided into reaches, and the 2-y flood (or alternatively the bank-full discharge) is estimated. In certain cases it may be desirable to use discharges corresponding to 10-y frequencies or more. The mean velocity is computed, and the reach concentration time is calculated by using the reach valley length (straight distance). The sum of the concentration time for all reaches is the concentration time for the catchment. The lag is estimated as follows:

tl        6 ___ = ____  tc       10

(5-32)

in which t_l = lag and t_c = concentration time NRCS experience has shown that this ratio is typical of midsize catchments [21].

tp ___ = 5  tr

(5-33)

which is close to Snyder's ratio of 6. Assuming uniform effective rainfall for simplicity, the time-to-peak is by definition equal to

tr tp = ____ + tl          2

(5-34)

Eliminating t_r from Eqs. 5-33 and 5-34, leads to

tp      10 ___ = ____  tl         9

(5-35)

Therefore:

tr        2 ___ = ____  tl        9

(5-36)

and

tr        2 ___ = ____  tl        15

(5-37)

To derive the NRCS unit hydrograph peak flow formula, the ratio Tb/tp = 8/3 is used in Eq. 5-18, leading to

(3/4)A tp = _________             tp

(5-38)

In SI units, the peak flow formula is:

2.08A tp = _________             tp

(5-39)

in which Q_p = unit hydrograph peak flow for 1 cm of effective rainfall in cubic meters per second; A = catchment area in s uare kilometers; and t_p = time-to-peak in hours. In U.S. customary units, the NRCS peak flow formula is:

484A tp = _________             tp

(5-40)

in which Q_p = unit hydrograph peak flow for 1 in. of effective rainfall; A = catchment area in square miles; and t_p = time-to-peak in hours.

Given Eqs. 5-32 and 5-34, the time-to-peak can be readily calculated as follows: t_p = 0.5t + 0.6t_c Once t_p and Q_p have been determined, th NRCS imensionless umt hydrograph (Fig. 5-12) 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 ( T_bt / t_p = 8/3 ) used to develop the

Figure 5-12  NRCS dimensionless unit hydrogrpah [21].

peak flow value. The dimensionless unit hydrograph has a value of T_b / t_p = 5. Values of NRCS dimensionless unit hydrograph ordinates at intervals of 0.2 (t / t_p) are given in Table 5-6.

The NRCS method provides a unit hydrograph shape and therefore leads to more reproducible results than the Snyder method. However, the ratio T_bt / t_p is kept constant and equal to 8/3. Also, when lag is calculated by the velocity method, the ratio t_l / t_c 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 T_bt / t_p other than 8/3 may lead to other shapes of unit hydrographs. Larger values of T_bt / t_p (equivalent to lower values of C_p in the Snyder method) imply greater catchment storage. Therefore, since the NRCS method fixes the value of T_bt / t_p it should be limited to midsize catchments in the lower end of the spectrum (2.5-250 km²). The Snyder method, however, by providing a variable Tb,ltp, may be used for larger catchments (from 250 up to 5000 km²) [10].

Efforts to extend the range of applicability of the NRCS method have led to the relaxation of the T_bt / t_p ratio. It can be shown that the ratio p of volume-to-peak (volume under the rising limb of the triangular unit hydrograph) to the triangular unit hydrograph volume is the reciprocal of the ratio T_bt / t_p. For instance, in the case of the standard NRCS Synthetic unit hydrograph, T_bt / t_p = 8/3, and p = 3/8. In terms of p , Eq. 5-38 can be expressed as follows:

2pA tp = _______            tp

(5-41)

Table 5-6 NRCS Dimensionless Unit Hydrograph Ordinates t / tp Q / Qp t / tp Q / Qp t / tp Q / Qp t / tp Q / Qp t / tp Q / Qp 0.0 0.00 0.2 0.10 1.2 0.93 2.2 0.207 3.2 0.040 4.2 0.0100 0.4 0.31 1.4 0.78 2.4 0.147 3.4 0.029 4.4 0.0070 0.6 0.66 1.6 0.56 2.6 0.107 3.6 0.021 4.6 0.0030 0.8 0.93 1.8 0.39 2.8 0.077 3.8 0.015 4.8 0.0015 1.0 1.00 2.0 0.28 3.0 0.055 4.0 0.011 5.0 0.0000

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:

1. Catchment lag,

2. Ratio of triangular time base to time-to-peak, and

3. Ratio of actual time base to time-to-peak.

In addition, the NRCSmethod specifies a gamma function for the shape of the unit hydrograph. Many other synthetic unit hydrographs have been reported in the literature [16]. In general , any procedure defining geometric properties and hydrograph shape can be used to develop a synthetic unit hydrograph.

Example 5-3. Calculate the NRCS synthetic unit hydrograph for a 6.42 km2 catchment with the following data: Hydraulic length L = 2204 m; runoff curve number CN = 62; average land slope Y = 0.02. Using Eq. 5-30, tl = 1.8 h. Therefore: tr = 0.4 h; tp = 2 h; Tb = 10 h. Using Eq. 5-39, Qp = 6.68 m3/ s. Using Table 5-6, the ordinates of the unit hydrograph are calculated as shown in Table 5-7.

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 X-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:

1. The superposition method and

2. The S-hydrograph method.

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

Table 5-7 Unit Hydrograph Ordinates: Example 5-4 ( Qp = 6.68m3; tp = 2h ) t / tp t(h) t / tp Q (m3/s) 0.0 0.00 0.0 0.000 0.2 0.10 0.4 0.668 0.4 0.31 0.8 2.071 0.6 0.66 1.2 4.410 0.8 0.93 1.6 6.212 1.0 1.00 2.0 6.680 1.2 0.93 2.4 6.212 1.4 0.78 2.8 6.212 1.6 0.56 3.2 3.740 1.8 0.39 3.6 2.605 2.0 0.28 4.0 1.870 2.2 0.207 4.4 1.382 2.4 0.147 4.8 0.982 2.6 0.107 5.2 0.714 2.8 0.077 5.6 0.514 3.0 0.055 6.0 0.367 3.2 0.040 6.4 0.267 3.4 0.029 6.8 0.194 3.6 0.021 7.2 0.140 3.8 0.015 7.6 0.100 4.0 0.011 8.0 0.073 4.2 0.010 8.4 0.067 4.4 0.007 8.8 0.047 4.6 0.003 9.2 0.020 4.8 0.0015 9.6 0.010 5.0 0.0000 10.00 0.000

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 T_b is the time base of the X-hour hydrograph, the time base of the nX-hour hydrograph is equal to T_b + (n - l)X. The procedure is illustrated by the following example.

Example 5-5. Use the superposition method to calculate the 2-h and 3-h unit hydrographs of a catchment, based on the following I-h unit hydrograph: Time (h) 0 1 2 3 4 5 6 7 8 9 10 11 12 Flow (m3/s) 0 100 200 400 800 700 600 500 400 300 200 100 0 The calculations are shown in Table 5-8. Column 1 shows the time in hours. Column 2 shows the ordinates of the 1-h unit hydrograph. Column 3 shows the ordinates of the 1-h unit hydrograph, lagged 1 h. Column 4 shows the ordinates of the I-h unit hydrograph, lagged 2 h. Column 5 shows the ordinates of the 2-h unit hydrograph, obtained by summing the ordinates of Cols. 2 and 3 and dividing by 2. Column 6 shows the ordinates of the 3-h unit hydrograph, obtained by summing the ordinates of Cols. 2, 3, and 4, and dividing by 3. The sum of ordinates for I-h, 2-h, and 3-h unit hydrographs is the same: Table 5-8 Change in Unit Hydrograph Duration, Superposition Method: Example 5-5. (1) (2) (3) (4) (5) (6) Time (h) 1-h UH Lagged 1 h Lagged 2 h 2-h UH 3-h UH (m3/s) 0 0 0 0 0 0 1 100 0 0 50 33 2 200 100 0 150 100 3 400 200 100 300 233 4 800 400 200 600 467 5 700 800 400 750 633 6 600 700 800 650 700 7 500 600 700 550 600 8 400 500 600 450 500 9 300 400 500 350 400 10 200 300 400 250 300 11 100 200 300 150 200 12 0 100 200 50 100 13 0 0 100 0 33 14 0 0 0 0 0 Sum 4300 4300 4299 4300 m3 / s. The time base of the láh unit hydrograph is 12 h, whereas the time base of the 2áh unit hydrograph is 13 h and the time base of the 3áh unit hydrograph is 14 h.

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:

1. Determine the X-hour S-hydrograph (Fig. 5-13).

2. The X-hour S-hydrograph is derived by accumulating the unit hydrograph ordinates at intervals equal to X.

3. Lag the X-hour S-hydrograph by a time interval equal to Y hours.

4. Subtract ordinates of the two previous S-hydrographs.

5. Multiply the resulting hydrograph ordinates by X / Y to obtain the Y-hour unit hydrograph.

The volume under X-hour and Y-hour unit hydrographs is the same. If T_b is the time base of the X-hour unit hydrograph, the time base of the Y-hour unit hydrograph is T_b - X + Y.

Example 5-6. For the 2-h unit hydrograph calculated in the previous example, derive the 3-h unit hydrograph by the S-hydrograph method. Use this 3-h unit hydrograph to derive the 2-h unit hydrograph, confirming the applicability of the S-hydrograph method, regardless of the ratio between X and Y. The calculations are shown in Table 5-9. Column 1 shows the time in hours. Column 2 shows the 2-h unit hydrograph ordinates calculated in the previous example. Figure 5-13  Sketch of unit hydrograph and corresponding S-hydrograph. Column 3 is the 2-h S-hydrograph, obtained by accumulating the ordinates of Col.2 at intervals of X = 2 h. Column 4 is the S-hydrograph of Col. 3 lagged Y = 3 h. Column 5 is equal to Col. 3 minus Col. 4. Column 6 is the product of Col. 5 times X / Y = 2/3 Column 6 is the 3-h unit hydrograph.Its sum is 4299 m3/ s, the same as the sum of Col. 2, confirming that is contains a unit volume. The time base of the 2-h unit hydrograph is 13 h, and the time base of the 3-h unit hydrograph is 14 h. Column 7 is the 3-h S-hydrograph, obtained by accumulating the ordinates of Col. 6 at intervals of X = 3 h. Column 8 is the S-hydrograph of Col. 7 lagged Y = 2 h. Column 9 is equal to Col. 7 minus Col. 8. Column 10 is the product of Col. 9 times X / Y = 3/2. Column 10 is the 2-h unit hydrograph, and it is confirmed to be the same as that of Col. 2.

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. 5-13).

Convolution and Composite Hydrographs

The procedure to derive a composite or flood hydrograph based on a unit hydro graph 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 T_b 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 T_b - X + nX = T_b + (n - 1)X. The convolution procedure is illustrated by the following example.

Example 5-7. Assume that the following 1-h unit hydrograph has been derived for a certain watershed: Time (h) 0 1 2 3 4 5 6 7 8 9 Flow (m3/s) 0 100 200 400 800 600 400 200 100 0 Table 5-9 Change in Unit Hydrograph Duration, S-Hydrograph Method: Example 5-6. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Time (h) 2-h UH 2-h SH Lagged 3 h Col.3 - Col.4 3-h UH 3-h SH Lagged 2 h Col.7 - Col.8 2-h UH 0 0 0 0 0 0 0 0 0 0 1 50 50 0 50 33 33 0 33 50 2 150 150 0 150 100 100 0 100 150 3 300 350 0 350 233 233 33 200 300 4 600 750 50 700 467 500 100 400 600 5 750 1100 150 950 633 733 233 500 750 6 650 1400 350 1050 700 933 500 433 650 7 550 1650 750 900 600 1100 733 367 550 8 450 1850 1100 750 500 1233 933 300 450 9 350 2000 1400 600 400 1333 1100 233 350 10 250 2100 1650 450 300 1400 1233 167 250 11 150 2150 1850 300 300 1433 1333 100 150 12 50 2150 2000 150 100 1433 1400 33 50 13 0 2150 2000 150 100 1433 1433 0 0 14 0 2150 2150 0 0 1433 1433 0 0 Sum 4300 4299 4300 A 6-h storm with a total of 5 cm of effective rainfall covers the entire watershed and is distributed in time as follows: Time (h) 0 1 2 3 4 5 6 Effective rainfall (cm) 0.1 0.8 1.6 1.2 0.9 0.4 Calculate the composite hydrograph using the convolution technique. The calculations are shown in Table 5-10. Column 1 shows the time in hours, and Col. 2 shows the unit hydrograph ordinates in cubic meters per second. Column 3 shows the product of the first-hour rainfall depth times the unit hydrograph ordinates. Column 4 shows the product of the second-hour rainfall depth times the unit hydrograph ordinates, lagged 1 h with respect to Col. 3.The computational pattern established by Cols. 3 and 4 is the same for Cols. 5-8. Column 9, the sum of Cols. 3 through 8, is the composite hydrograph for the given storm pattern. The sum of Col. 2 is 2800 m3/ s and is equivalent to 1 cm of net rainfall. The sum of Col. 9 is verified to be 14,000 m3/s, and, therefore, the equivalent of 5 cm of effective rainfall. The time base of the composite hydrograph is Tb = 9 + (6 - 1) X 1 = 14 h.

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 5-10).

Table 5-10 Composite Hydrograph by Convolution: Example 5-6. (1) (2) (3) (4) (5) (6) (7) (8) (9) Time (h) UH ( m3/s) 0.1 X UH 0.8 X UH 1.6 X UH 1.2 X UH 0.9 X UH 0.4 X UH Composite Hydrograph 0 0 0 __ __ __ __ __ 0 1 100 10 0 __ __ __ __ 10 2 200 20 80 0 __ __ __ 100 3 400 40 160 160 0 __ __ 360 4 800 800 320 320 120 0 __ 840 5 600 60 640 640 240 90 0 1670 6 400 40 480 1280 480 180 40 2500 7 200 20 320 960 960 360 80 2700 8 100 10 160 640 720 720 160 2410 9 0 0 80 320 480 540 320 1740 10 __ __ 0 160 240 360 240 1000 11 __ __ __ 0 120 180 160 460 12 __ __ __ __ 0 90 80 170 13 __ __ __ __ __ 0 40 40 14 __ __ __ __ __ __ 0 0 Sum 2800 14,000

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:

N = m + n - 1

(5-42)

Therefore:

m = N - n + 1

(5-43)

By elimination and back substitution, the following formula can be developed for the unit hydrograph ordinates u_i as a function of storm hydrograph ordinates q_i and effectiverainfall depths r_k :

k = 2,n         qi  __   Σ          uj rk                   j = i - 1.1 ui  =  ___________________                   r1

(5-44)

for i varying from 1 to m. In the summation term, j decreases from i-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 [12]. Other methods to derive unit hydrographs for complex storms are discussed by Singh [16].

Example 5-8. Use Eq. 5-44 and the storm hydrograph obtained in the:previous example to calculate the unit hydrograph. Since N = 13 and n = 6: m = 8. The first ordinate is u1 = q1 / r1 = 10/ 0.1 = 100. The second ordinate is u2 = (q2 - u1r2) = (100 - 100 X 0.8)/0.1 = 200. The third ordinate is u3 = [q3 - (u2r2 + u1r3)] / r1 = [360 - (200 X 0.8 + 100 X 1.6)]10.1 = 400. The fourth ordinate is u4 = [q4 - (u3r2 + u2r3 + u1r4) ] / r1 = [840 - (400 X 0.8 + 200 X 1.6 + 100 X 1.2)]/0.1 = 800. The remaining ordinates are obtained in a similar way.

5.4  NRCS TR-55 METHOD

[Questions]   [Problems]   [References]   •   [Top]   [Runoff Curve Number]   [Unit Hydrograph]

The TR-55 method is a collection of simplified procedures developed by the USDA Natural Resources Conservation Service (formerly Soil Conservation Service) to calculate peak discharges, storm hydrographs, and stormwater storage volumes in small/midsize urban catchments [22]. It consists of three methodologies:

1. a graphical method for flood peak discharge determination,

2. a tabular method for hydrograph computation, and

3. a detention-basin method to size stormwater storage facilities.

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 [22]. 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 lA, type II, and type III (Fig. 5-14). Type I appltes 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 (FIg. 5-15 and Chapter 13).

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 2 h are contained within the NRCS distributions . For instance, if a lO-y 24-h rainfall distribution is used, the 1-h period with the most intense rainfall corresponds to the 10-y 1-h rainfall depth.

TR-55 uses the runoff curve number method (Section 5.1) to abstract total rainfall depth and calculate runoff depth. The abstraction procedure follows the

Figure 5-14  NRCS 24-h rainfall distribution [22].

guidelines established in NEH-4 [21], with extensions to account for curve numbers applicable to urban areas. In addition, TR-55 includes procedures to determine time of concentration for the following types of surface flow:

1. Overland flow,

2. Shallow concentrated flow, and

3. Streamflow.

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 hydro graph. 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.

Figure 5-15  Approximate geographical boundaries for NRCS rainfall distribution [22].

Selection of Runoff Curve Number

To estimate curve numbers for urban catchments, TR-55 defines two types of areas:

1. Pervious, and

2. Impervious.

Once pervious and impervious areas are delineated, the percent imperviousness can be determined.

Impervious areas are of two kinds:

1. Connected, and

2. Unconnected.

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

1. If runoff from it flows directly into the drainage system, or

2. If runoff from it occurs as shallow concentrated flow which runs first over a pervious area and then into a drainage system.

An impervious area is considered unconnected if runoff from it spreads over a pervious area as overland (sheet) flow.

Table 5-2(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:

1. Impervious areas are directly connected to the drainage system and have a CN = 98; and

2. Pervious areas are considered equivalent to pasture (open space in Table 5-2(a)) in good hydrologic condition.

Tables 5-2(b), (c), and (d) show runoff curve numbers for cultivated agricultural lands, other agricultural lands, and arid and semiarid rangelands, respectively.

Figure 5-16 is used in lieu of Table 5-2(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 5-2(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 5-2(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 Figure 5-16 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 5-17 is used to determine a composite eN 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. 5-16 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 Figure 5-17, 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. 5-17) 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:

0.007 ( nL )0.8 tt = ______________             p20.5S0.4

(5-45)

in which t_t, = travel time in hours; n = Manning n; L = flow length, in feet;

Figure 5-16  Composite CN as a function of impervious area percent and pervious area CN [22].

Figure 5-17  Composite CN as a function of total impervious area percent, ratio of unconnected impervious area to total impervious area, and pervious area CN [22].

p₂ = 2-y 24-h rainfall depth in inches; and S = average land slope, in feet per foot. In SI units, this equation is:

0.0288 ( nL )0.8 tt = ______________             p20.5S0.4

(5-46)

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 5-11.

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. 5-18. For streamflow, the Manning equation (Eq. 2-89) 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].

Table 5-11 TR-55 Manning n Values for Overland Flow [22]. Surface Description Manning n Smooth surfaces   (concrete, asphalt, gravel, or bare soil) 0.011 Fallow (no residue) 0.05 Cultivated ground   (residue civer less than or equal to 20%) 0.06   (residue cover greater than 20%) 0.17 Grass   Short Prarie 0.15   Dense 0.24   Bermuda 0.41   Range (natural) 0.13 Woods   Light underbrush 0.40   Dense underbrush 0.80 Note: Dense grass includes weeping lovegrass, bluegrass, buffalo grass, blue gamma grass, native grass mixture, alfalfa, and the like.

Figure 5-18  Average velocitites for estimating travel time for shallow concentrated flow [22].

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 flower unit area, per unit runoff depth. In TR-55, unit peak flow is a function of

1. Time of concentration

2. Ratio of initial abstraction to total rainfall, and

3. storm type.

Peak discharge is calculated by the following formula:

Qp = quAQF

(5-47)

in which Q_p = peak discharge in L³T^-1 units; q_u = 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 overland flow , shallow concentrated flow , or streamflow. The runoff curve number is determined from either Table 5-2. Fig. 5-16, or Fig. 5-17. 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, Eqs. 5-8, or 5-9.

The initial abstraction is calculated by combining Eqs. 5-4 and 5-7 to yield:

200 Ia = ______ - 2          CN

(5-48)

in which I_a = initial abstraction, in inches. The equivalent SI formula is:

508 Ia = ______ - 5.08          CN

(5-49)

in which I_a is given in centimeters.

The surface storage correction factor F is obtained from Table 5-12 as a function of the percentage of pond and swamp areas. With time of concentration t_c, ratio I_a / P, and storm type (either I, lA, II, or III). Fig. 5-19 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 I_a / P different than those shown in Fig. 5-19. For values of I_a / P outside of the range shown in Fig. 5-19, the maximum (or minimum) value should be used. To obtain unit peak flow in cubic meters per second per square kilometer per centimeter, the unit peak flow values obtained from Fig. 5-19 are multiplied by 0.0043. 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 10.0 h, and surface storage areas spread throughout the catchment and covering less than 5 percent of it.

Table 5-12 TR-55 Surface Storage Correction Factor F [22]. Percentage of Pond and Swamp Areas F 0.0 1.00 0.2 0.97 1.0 0.87 3.0 0.75 5.0 0.72 Note: Pond and swamp areas should be spread throughout the catchment

Example 5-9. Calculate the 10-y peak flow by the TR-55 graphical method using the following data: catchment area 4 km2 ; total impervious area 0.8 km2; unconnected impervious area 0.6 km2 ; pervious area curve number CN = 70; storm type II ; time of concentration 1.5 h; 10-y rainfall P = 9 cm; percentage of pond and swamp areas, 1 percent. Since there are unconnected impervious areas and the total impervious area amounts to less than 30 percent of the catchment, Fig. 5-17 is used to calculate the composite curve number. With total impervious area (20 percent), ratio of unconnected impervious to total impervious (0. 75), and pervious CN (70) areas, the composite curve number from Fig. 5- 17 is CN = 74. The runoff depth Eq . 5-9) is Q = 3.23 cm. The initial abstraction (Eq. 5-49) is Ia = 1.78 cm, and the ratio Ia / P = 0.2. From Fig. 5-19(c) (storm type II), time of concentation 1.5 h, and Ia / P = 0.2 , the unit peak flow is 250 ft3/ (s-mi2-in.) or 250 X 0.0043 = 1.075 m3/ (s-km2-cm). From Table 5-12, F = 0.87. From Eq. 5-47, with qu = 1.075 m3/ (s-km2-cm); A = 4 km2; Q = 3.23 cm; and F = 0.87, the peak discharge is Qp = 12.08 m3/s.

Example 5-10. Calculate the 25-y peak flow by the TR-55 graphical method using the following data: urban watershed of area A = 1.5 mi2 ; surface flow is shallow concentrated, paved, hydraulic length L = 4,320 ft; slope S = 0.014; 26 percent of the watershed is 1/3-acre lots, 30 percent impervious, hydrologic soil group B; 42 percent of the watershed is 1/2-acre lots, with lawns in fair hydrologic condition, 36 percent impervious, hydrologic soil group C; 32 percent of the watershed is l/2-acre lots, with lawns in good hydrologic condition, 24 percent total impervious, 50 percent unconnected, hydrologic soil group C; storm type I; 25-y rainfall P = 5 in., and 0.2 percent pond and swamps areas. From Fig. 5-18, the average velocity along the hydraulic length is v = 2.4 ft/s; therefore, the time of concentration is tc = L / v = 0.5 h. For the 26 percent subarea, with 1/3-acre lots, 30 percent impervious, the curve number is obtained directly from Table 5-2(a): CN = 72. For the 42 percent subarea, with 1/2-acre lots, 36 percent impervious, first the pervious area CN is obtained from Table 5-2(a) (open space in fair hydrologic condition, soil group C): CN = 79; then, the composite CN is obtained from Fig. 5-16: CN = 86. For the 32 percent subarea, with 1/2- acre lots, 24 percent total impervious, 50 percent unconnected, first the pervious area CN is obtained from Table 5-2 (a) (open space in good hydrologic condition, soil group C): CN = 79; then, the composite eN is obtained from Fig. 5-17: CN = 78. The composite eN for the entire watershed is: CN = (0.26 X 72) + (0.42 X 86) + (0.32 X 78) = 80. The runoff depth (Eq. 5.8) is: Q = 2.9 in. The initial abstraction (Eq. 5-48) is: Ia = 0.5 in.; then, the ratio Ia = 0.1. The unit peak: flow (Fig. 5-19(a)) is: qu = 282 ft3/(s-mi2-in.). The surface storage correction factor (Table 5-12) is: F = 0.97. Finally, the peak: flow (Eq. 5-47) is: Qp = 282 X 1.5 X 2.9 X 0.97 = 1190 ft3/s.

Figure 5-19   Unit peak discharge in TR-55 graphical method:(a)NRCS type I rainfall distribution; (b) NRCS type IA rainfall distribution; (c) NRCS type II rainfall distribution; (d)NRCS type III rainfall distribution [22].

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 tim of concentration ,abstraction parameter I_a/P, and 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. 5-19, it is 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 I_a/P is related to the catchment's abstractive properties. The greater the curve number, the lesser the value of I_a/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 locaation 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 can 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 time of concentration to 10 h.

The unit values of catchment area, runoff depth, and time of concentration can be used to provide a comparison between the TR-55 graphical method and the rationa method. To illustrate, assume a catchment area of 1 mi² (640 ac), time of concentration 1 h, and 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 Q_p= 613ft³/ s.

A calculation with the TR-55 graphical method, using the lowest possible value of abstraction for comparison purposes (I_a/P = 0.10), gives the following: For storm type 1, 200 ft³/s; type lA, 108 ft³/s; type 11,360 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-5S graphical method generally gives lower peak flows than the rational method. This may be attributed to the fact that the TR-S5 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 comprable, since the value of I_a/P = 0.1 does not correspond exactly to C = 0.95.

QUESTIONS

[Problems]   [References]   •   [Top]   [Runoff Curve Number]   [Unit Hydrograph]   [TR-55 Method]

1. What catchment properties are used in estimating a runoff curve number? What significant rainfall characteristic is absent from the NRCS runoff curve number method?

2. What is the antecedent moisture condition in the runoff curve number method? How is it estimated?

3. What is hydrologic condition in the runoff curve number method? How is it estimated?

4. Describe the procedure to estimate runoff curve numbers from measured data. What level of antecedent moisture condition will cause the greatest runoff? Why?

5. What is a unit hydrograph? What does the word unit refer to?

6. Discuss the concepts of linearity and superposition in connection with unit hydrograph theory.

7. What is catchment lag? Why is it important in connection with the calculation of synthetic unit hydrographs?

8. In the Snyder method of synthetic unit hydrographs, what do the parameters C_i, and C_p describe?

9. Compare lag, time-to-peak, time base, and unit hydrograph duration in the Snyder and NRCS synthetic unit hydrograph methods.

10. What is the shape of the triangle used to develop the peak flow formula in the NRCS synthetic unit hydrograph method? What value of Snyder's C_p matches the NRCS unit hydrograph?

11. What elements are needed to properly define a synthetic unit hydrograph?

12. What is the difference between superposition and S-hydrograph methods to change unit hydrograph duration? In developing S-hydrographs, why are the ordinates summed up only at intervals equal to the unit hydrograph duration?

13. What is hydrograph convolution? What assumptions are crucial to the convolution procedure?

14. What is an unconnected impervious area in the TR-SS methodology? What is unit peak flow?

15. Given the similarities between the TR-SS graphical method and the rational method, why is the former based on runoff depth while the latter is based on rainfall intensity?

PROBLEMS

[References]   •   [Top]   [Runoff Curve Number]   [Unit Hydrograph]   [TR-55 Method]   [Questions]

1. An agricultural watershed has the following hydrologic characteristics: (1) a subarea in fallow , with bare soil, soil group B, covering 32 0/0; and (2) a subarea planted with row crops, contoured and terraced, in good hydrologic condition, soil group C, covering 68%. Determine the runoff Q, in centimeters, for a 10.5-cm rainfall. Assume an AMC II antecedent moisture condition.

2. A rural watershed has the following hydrologic characteristics:

   1. (1) a pasture area, in fair hydrologic condition, soil group B, covering 22%;

   2. a meadow, soil group B, covering SS% ; and

   3. woods, poor hydrologic condition, soil group B-C, covering 23%.

   Determine the runoff Q, in centimeters, for a 12-cm rainfall. Assume an AMC III antecedent moisture condition.

3. Rain falls on a 9.5-ha urban catchment with an average intensity of 2.1 cm/h and duration of 3h. The catchment is divided into (1) business district (with 85% impervious area), soil group C, covering 20%; and (2) residential district, with t-ac average lot size (with 30% impervious area), soil group C. Determine the tntal runoff volume, in cubic meters, assuming an AMC II antecedent moisture condition.

4. Rain falls on a 950-ha catchment in a semiarid region. The vegetation is desert shrub in fair hydrologic condition. The soils are: 15% soil group A; 55% soil group B, and 30% soil group C. Calculate the runoff Q, in centimeters, caused by a 15-cm storm on a wet antecedent moisture condition. Assume that field data support the use of an initial abstraction parameter λ= 0.3.

5. The hydrologic response of a certain 10-mi² agricultural watershed can be modeled as a triangular-shaped hydrograph, with peak flow and time base defining the triangle. Five events encompassing a wide range of antecedent moisture conditions are selected for analysis. Rainfall-runoff data for these five events are as follows:

   Rainfall P (in.) Peak Flow Qp (ft3/s) Time base (h) 7.05 3100 12. 6.41 3700 14. 5.13 4100 13. 5.82 4500 12. 6.77 3500 14.

   
   Determine a value of AMC II runoff curve number based on the above data.

6. The following rainfall-runoff data were measured in a certain watershed:

   Rainfall P (cm) RunoffQp (cm) 15.2 12.3 10.5 10.1 7.2 4.3 8.4 5.2 11.9 9.1

   
   Assuming that the data encompass a wide range of antecedent moisture conditions, estimate the AMC II runoff curve number.

7. The following rainfall distribution was observed during a 6-h storm:

   Time (h) 0 2 4 6 Intensity (mm/h) 10 15 12

   
   The runoff curve number is CN = 76. Calculate the φ-index.

8. The following rainfall distribution was observed during a 12-h storm:

   Time (h) 0 2 4 6 8 10 12 Intensity (mm/h) 5 10 13 18 3 10

   
   The runoff curve number is CN = 86. Calculate the φ-index.

9. The following rainfall distribution was observed during a 6-h storm:

   Time (h) 0 2 4 6 Intensity (mm/h) 18 24 12

   
   The φ-index is 10 mm/ h. Calculate the runoff curve number.

10. The following rainfall distribution was observed during a 24-h storm:

    Time (h) 0 3 6 9 12 15 18 21 24 Intensity (mm/h) 5 8 10 12 15 5 3 6

    
    The φ-index is 4 mm/h. Calculate the runoff curve number.

11. A unit hydrograph is to be developed for a 29.6-km² catchment with a 4-h T2 lag. A 1-h rainfall has produced the following runoff data:

    Time (h) 0 1 2 3 4 5 6 7 8 9 10 11 12 Flow (m3/s) 1 2 4 8 12 8 7 6 5 4 3 2 1

    
    Based on this data, develop a 1-h unit hydrograph for this catchment. Assume baseflow is 1 m³/ s.

12. A unit hydrograph is to be developed for a 190.8-km² catchment with a 12-h T2 lag. A 3-h rainfall has produced the following runoff data:

    Time (h) 0 3 6 9 12 15 18 21 24 Flow (m3/s) 15 20 55 80 60 48 32 20 15

    
    Based on this data, develop a 3-h unit hydrograph for this catchment. Assume baseflow is 15 m³/s.

13. Calculate a set of Snyder synthetic unit hydrograph parameters for the following data: catchment area A = 480 km²; L = 28 km; L_c = 16 km; C_t = 1.45; and C_p = 0.61.

14. Calculate a set of Snyder synthetic unit hydrograph parameters for the following data: catchment area A = 950 km²; L = 48 km; L_c = 21 km; C_t = 1.65; and C_p = 0.57.

15. Calculate an NRCS synthetic unit hydrograph for the following data: catchment area A = 7.2 km² ; runoff curve number CN = 76; hydraulic length L = 3.8 km; and average land slope Y = 0.012.

16. Calculate an NRCS synthetic unit hydrograph for the following data: catchment area (natural catchment) A = 48 km²; runoff curve number CN = 80; hydraulic length L = 9 km; and mean velocity along hydraulic length V = 0.25 m / s.

17. Calculate the peak flow of a triangular SI unit hydrograph (1 cm of runoff) having a volume-to-peak to unit-volume ratio p = 3/10. Assume basin area A = 100 km², and time to- peak t_p = 6 h.

18. Given the following 1-h unit hydrograph for a certain catchment. find the 2-h unit hydrograph using (a) the superposition method and (b) the S-hydrograph method.

    Time (h) 0 1 2 3 4 5 6 Flow (ft3/s) 0 500 1000 750 500 250 0

    
    

19. Given the following 3-h unit hydrograph for a certain catchment. find the 6-h unit hydrograph using (a) the superposition method and (b) the S-hydrograph method.

    Time (h) 0 3 6 9 12 15 18 21 24 Flow (m3/s) 0 5 15 30 25 20 10 5 0

    
    

20. Given the following 2-h unit hydrograph for a certain catchment, find the 3-h unit hydrograph. Using this 3-h unit hydrograph, calculate the 1-h unit hydrograph.

    Time (h) 0 1 2 3 4 5 6 7 Flow (m3/s) 0 25 75 87.5 62.5 37.5 12.5 0

    
    

21. Given the following 4-h unit hydrograph for a certain catchment, find the 6-h unit hydrograph. Using this 6-h unit hydrograph, calculate the 4-h unit hydrograph, verifying the computations.

    Time (h) 0 2 4 6 8 10 12 14 16 18 20 22 24 Flow (m3/s) 0 10 30 60 100 90 80 70 50 40 20 10 0

    
    

22. Given the following 4-h unit hydrograph for a certain catchment: (a) Find the 6-h unit hydrograph; (b) using the 6-h unit hydrograph, calculate the 8-h unit hydrograph; (c) using the 8-h unit hydrograph, calculate the 4-h unit hydrograph, verifying the computations.

    Time (h) 0 2 4 6 8 10 12 14 16 18 20 Flow (m3/s) 0 10 25 40 50 40 30 20 10 5 0

    
    

23. The following 2-h unit hydrograph has been developed for a certain catchment:

    Time (h) 0 2 4 6 8 10 12 Flow (m3/s) 0 100 200 150 100 50 0

    
    A 6-h storm covers the entire catchment and is distributed in time as follows:

    Time (h) 0 2 4 6 Total rainfall (in./h) 1.0 1.5 0.5

    
    Calculate the composite hydrograph for the effective storm pattern, assuming a runoff curve number CN = 80.

24. The following 3-h unit hydrograph has been developed for a certain catchment:

    Time (h) 0 3 6 9 12 15 18 21 24 Flow (m3/s) 0 10 20 30 25 20 15 10 0

    
    A 12-h storm covers the entire catchment and is distributed in time as follows:

    Time (h) 0 3 6 9 12 Total rainfall (in./h) 6 10 18 2

    
    Calculate the composite hydrograph for the effective storm pattern, assuming a runoff curve number CN = 80.

25. A certain basin has the following 2-h unit hydrograph:

    Time (h) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Flow (m3/s) 0 5 15 30 60 75 65 55 45 35 25 15 5 0

    
    Calculate the flood hydrograph for the following effective rainfall hyetograph:

    Time (h) 0 3 6 Effective rainfall (cm/h) 1.0 2.0

    
    

26. Given the following flood hydrograph and effective storm pattern, calculate the unit hydrograph ordinates by the method of forward substitution.

    Time (h) 0 1 2 3 4 5 6 7 8 9 10 11 12 Flow (m3/s) 0 5 18 46 74 93 91 73 47 23 9 2 0

    
    

    Time (h) 0 1 2 3 4 6 6 Effective rainfall (cm/h) 0.5 0.8 1.0 0.7 0.5 0.2

    
    

27. Using TR-55 procedures, calculate the time of concentration for a watershed having the following characteristics:

    1. overland flow, dense grass, length L = 100 ft, slope S = 0.01 , 2-y 24-h rainfall P₂ = 3.6 in.;

    2. shallow concentrated flow, unpaved, length L = 1400 ft, slope S = 0.01; and

    3. streamflow, Manning n = 0.05, flow area A = 27 ft², wetted perimeter P = 28.2 ft, slope S = 0.005, length L = 7300 ft.

28. Using TR-55 procedures, calculate the time of concentration for a watershed having the following characteristics: (1) overland flow, bermuda grass, length L = 50 m, slope S = 0.02, 2-y 24-h rainfall P₂ = 9 cm; and (2) streamflow, Manning n = 0.05, flow area A =4.05 m², wetted perimeter P = 8.1 m, slope S = 0.01, length L = 465 m.

29. A 250-ac watershed has ttc following hydrologic soil-cover complexes:

    1. soil group B, 75 ac, urban, 1/2-ac lots with lawns in good hydrologic condition, 25% connected impervious;

    2. soil group C, 100 ac, urban, 1/2-ac lots with lawns in good hydrologic condition, 25% connected impervious; and

    3. soil group C, 75 ac, open space in good condition.
    
    Determine the composite runoff curve number.

30. A 120-ha watershed has the following hydrologic soil-cover complexes:

    1. soil group B, 40 ha, urban, 1/2-ac lots with lawns in good hydrologic condition, 35% connected impervious;

    2. soil group C, 55 ha, urban, 1/2-ac lots with lawns in good hydrologic condition, 35% connected impervious; and

    3. soil group C, 25 ha, open space in fair condition.
    
    Determine the composite runoff curve number.

31. A 90-ha watershed has the following hydrologic soil-cover complexes:

    1. soil group C, 18 ha, urban, 1/3-ac lots with lawns in good hydrologic condition, 30% connected impervious;

    2. soil group D, 42 ha, urban, 1/3-ac lots with lawns in good hydrologic condition, 40"70 connected impervious; and

    3. soil group D, 30 ha, urban, 1/3-ac lots with lawns in fair hydrologic condition, 30% total impervious, 25% of it unconnected impervious area.
    
    Determine the composite runoff curve number.

32. Use the TR-55 graphical method to compute the peak discharge for a 250-ac watershed, with 25-y 24-h rainfall P = 6 in., time of concentration t_c = 1.53 h, runoff curve number CN= 75, and Type II rainfall.

33. Use the TR-55 graphical method to calculate the peak discharge for a 960-ha catchment, with 50-y 24-h rainfall P = 10.5 cm, time of concentration t_c = 3.5 h, runoff curve number CN = 79, type I rainfall, and 1 % pond and swamp areas.

34. Calculate the 25-y peak flow by the TR-55 graphical method for the following watershed data:

    1. urban watershed, area A = 9.5 km²;

    2. surface flow is shallow concentrated, paved; hydraulic length L = 3850 m; slope S = 0.01;

    3. 42% of watershed is 1/3-ac lots, lawns with 85% grass cover, 34% total impervious, soil group C;

    4. 58% of the watershed is 1/3-ac lots, lawns with 95% grass cover, 24% total impervious, 25% of it unconnected, soil group C;

    5. Pacific Northwest region, 25-y 24-h rainfall P = 10 cm; 1% ponding.

35. Solve problem 5-32 using the computer program EH500 included in Appendix D.

36. Solve problem 5-33 using the computer program EH500 included in Appendix D.

REFERENCES

•   [Top]   [Runoff Curve Number]   [Unit Hydrograph]   [TR-55 Method]   [Questions]   [Problems]

1. Amorocho. J., and G. T. Orlob. (1961). "Nonlinear Analysis of Hydrologic Systems," University of California Water Resources Center, Contribution No. 40. November.

2. Barnes, H. H. Jr. (1967). "Roughness Characteristics of Natural Channels," U.S. Geological Survey Water Supply Paper No. 1849.

3. Chow, V. T. (1959). Open-Channel Hydraulics. New York: McGraw-Hill.

4. Diskin. M. H. (1964). "A Basic Study of the Nonlinearity of Rainfall-Runoff Processes in Watersheds," Ph.D. Diss., University of Illinois, Urbana.

5. Freeze, R. A., and J. A. Cherry. (1979). Groundwater, Englewood Cliffs, N.J.: Prentice- Hall.

6. French, R. H. (1986). Open-Channel Hydraulics, New York: McGraw-Hill.

7. Hall, M. J. (1984). Urban Hydrology. London: Elsevier Applied Science Publishers.

8. Hawkins, R. H., A. T. Hjelmfelt, and A. W. Zevenbergen. (1985). "Runoff Probability, Storm Depth, and Curve Numbers," Journal of the Irrigation and Drainage Division, ASCE, Vol. 111, No.4, December, pp. 330-340.

9. Hjelmfelt, A. T., K. A. Kramer, and R. E. Burwell. (1981). "Curve Numbers as Random Variables," Proceedings. International Symposium on Rainfall-Runoff Modeling, Mississippi State University, (also Water Resources Publications, Littleton, Colorado).

10. Linsley, R. K. , M. A. Kohler, and 1. L. H. Paulhus. (1962). Hydrology for Engineers, 3d. ed. New York: McGraw-Hill.

11. McCuen, R. H., W. 1. Rawls, and S. L. Wong. (1984). "SCS Urban Peak Flow Methods," Journal of Hydraulic Engineering, ASCE, Vol. 110, No.3, March, pp. 290-299.

12. Newton, D. J. , and J. W. Vineyard. (1967). "Computer-Determined Unit Hydrographs from Floods," Journal of the Hydraulics Division, ASCE, Vol. 93, No. HY5, pp. 219-236.

13. Rallison, R.E., and R. G. Cronshey. (1979). Discussion of "Runoff Curve Numbers with Varying Soil Moisture," Journal of the Irrigation and Drainage Division ASCE, Vol. l05, No. IR4, pp. 439-441.

14. Sherman, L. K. (1932). "Streamflow from Rainfall by Unit-Graph Method," Engineering News-Record, Vol. 108, April 7, pp. 501-505.

15. Singh, K. P. (1962). "A Nonlinear Approach to the Instantaneous Unit Hydrograph," Ph.D. Diss., University of Illinois, Urbana.

16. Singh, V. P. (1988). Hydrologic Systems. Vol. 1: Rainfall-Runoff Modeling. Englewood Cliffs, N.J.: Prentice-Hall.

17. Snyder, F. F. (1938). "Synthetic Unit-Graphs," Transactions, American Geophysical Union. Vol. 19, pp. 447-454.

18. Springer, E. P., B. J. McGurk, R. H. Hawkins, and G. B. Coltharp. (1980). "Curve Numbers from Watershed Data," Proceedings. Symposium on Watershed Management, ASCE, Boise, Idaho, July, pp. 938-950.

19. Taylor, A. B., and H. E. Schwarz. (1952). "Unit Hydrograph Lag and Peak Flow Related to Basin Characteristics," Transactions. American Geophysical Union. Vol. 33, pp. 235-246.

20. U.S. Army Corps of Engineers. (1959). "Flood Hydrograph Analysis and Computations," Engineering and Design Manual EM 1110-2-1405, Washington, D.C.

21. USDA Soil Conservation Service. (1985). SCS National Engineering Handbook. Section 4: Hydrology, Washington, D.C.

22. USDA Soil Conservation Service. (1986). "Urban Hydrology for Small Watersheds," Technical Release No. 55 (TR-55), Washington, D.C.

23. USDA Soil Conservation Service. (1993). "Chapter 4: Stonn Rainfall Depth," Part 630, Hydrologic Engineering, Washington, D.C. (part of SCS National Engineering Handbook, Section 4: Hydrology).

24. U.S. Forest Service. (1959). Forest and Range Hydrology Handbook, Washington, D.C.

25. Forest Service. (1959). Handbook on Methods of Hydrologic Analysis, Washington,D.C.

26. Van. Sickle, D. (1969). "Experience with the Evaluation of Urban Effects for Drainage Design," in Effects of Watershed Changes on Streamflow, Proceedings, Water Resources Symposium No.2, University of Texas, Austin, pp. 229-254.

Suggested Readings

1. Sherman, L. K. (1932). "Streamflow from Rainfall by Unit-Graph Method," Engineering News-Record. Vol. 108, April 7, pp. 501-505.

2. Snyder, F. F. (1938). "Synthetic Unit-Graphs," Transactions. American Geophysical Union. Vol. 19, pp. 447-454.

3. USDA Soil Conservation Service. (1985). SCS National Engineering Handbook, Section 4: Hydrology, Washington, D.C.

4. USDA Soil Conservation Service. (1986). "Urban Hydrology for Small Watersheds," Technical Release No. 55 (TR-55), Washington, D.C.