Hydrological Sciences - Journal — des Sciences Hydrologiques, 30, 3, 9/1985

The meaning of oscillations in unit hydrograph S-curves

BRUCE HUNT Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand

ABSTRACT The Duhamel superposition integral is used to obtain some exact solutions for unit hydrograph applica­tions. These equations and numerical examples are used to show that oscillations will occur in an S-curve when the time step is less than the excess rainfall duration if the measured hydrograph differs from a hydrograph that would be obtained by solving a linear differential equation with time-independent coefficients. The implications of this result with regard to the calculation of the instantaneous unit hydrograph (IUH) are discussed.

La signification des oscillations sur les courbe en S de 1'hydrogramme unitaire RESUME L'intégrale de superposition de Duhamel est employée pour obtenir des solutions exactes pour des applications de 1'hydrogramme unitaire. Ces équations et des exemples numériques sont utilisés pour démontrer que des oscillations se produisent dans une courbe en S quand le pas de temps est inférieur à la durée de la pluie excédentaire si 1'hydrogramme mesuré est différent de 1'hydrogramme qui serait obtenu par la résolution d'une équation différentielle linéaire à coefficients indépendants du temps. On discute les implications de ces résultats relatifs au calcul de 1'hydrogramme unitaire instantané (IUH).

INTRODUCTION

The unit hydrograph has been used extensively to predict flood flows in rivers since its introduction by Sherman (1932a,b) over 50 years ago. Unit hydrograph theory is founded upon the two assumptions of linearity and time invariance, and Dooge (1973) has pointed out more recently that these assumptions are equivalent to the use of super­position or convolution integrals in linear systems theory. Numerous hydrologists have noted the presence of oscillations that occur during the solution of various inverse problems associated with unit hydrograph applications. For example, Linsley et al. (1975) note the presence of oscillations in S-curves and give a plausible explanation for their occurrence. Oscillations in the instantaneous unit hydrograph (IUH) are described and discussed by Blank et al. (1971), and Hall (1977) suggests a method for smoothing oscillations in finite-period unit hydrographs that are derived from complex storms by the harmonic method. This study uses the Duhamel super­position integral to obtain some exact solutions for unit hydrograph

331

332 Bruce Hunt

applications. Then perturbations are introduced into those solutions, and conclusions are made about the cause and significance of the resulting oscillations in calculated S-curves.

THE SUPERPOSITION INTEGRAL

If a specified variable, x, and a dependent variable, y, appear linearly in a differential equation whose coefficients do not depend upon time, t, then standard arguments given in mathematics texts such as Hildebrand (1962) suggest that x and y are related to each other by the Duhamel superposition integral:

y(t) = x(0) S(t) + /* S(t - T) x'(T) dT (1)

An integration by parts leads to the following alternative form for equation (1):

r* - *. 3S(t - T ) ^ .„. y(t) = fQ x(T) ^ — dT (2)

The functions x, y and S in equations (1) and (2) also depend upon spatial coordinates if x and y appear in a partial differential equation rather than an ordinary differential equation. The functions y(t) and S(t) are assumed to vanish for t £ 0, and x(t) is assumed to vanish for t < 0. Thus, the function x(t) may, in general, be discontinuous as t approaches zero. Use of the identity:

(3) 3S(t - T ) = _ 3S(t - T)

3t 3T

in equation (2) shows that S(t) is the solution for y when:

x(t) = 0 for -°° < t < 0

= 1 for 0 < t < °° (4)

Equation (4) implies that S(t) will become asymptotic to a horizontal line as t -*- °° if a steady-state solution exists for y(t) as t + ». A plot of S(t) vs. t will be referred to as an S-curve.

Integrating both sides of equation (2) with respect to time gives:

Q y<t> dt = /0°° i l X(T) ̂ f ^ 2 dT dt (5)

Interchanging the order of integration on the right side of equation (5) gives:

/0 y(t) dt = / 0 x(T> / T - ^ dt dT (6)

Finally, changing the integration variable from t to t' with the transformation t. ' = t - T in the inside integral leads to the result:

The meaning of oscillations in S curves 333

- CO

'o 3 S ( t ) 0 CO » CO

dt = / y ( t ) d t / / x ( t ) dt (7)

in which all dummy integration variables have been replaced with t. Since S(0) = 0, equation (7) is also equivalent to:

CO CO

S(°°) = J* y(t) dt// x(t) dt (8)

Equations (7) and (8) assume that x(t), y(t) and 3S(t)/9t all vanish rapidly enough as t -> °° to ensure the existence of the integrals.

UNIT HYDROGRAPH THEORY

In unit hydrograph applications, it is advantageous to set y(t) = Q(t) and x(t) = AI(t) in which Q(t) = direct runoff rate past a particular downstream cross section, A = basin area upstream of the cross section where Q is measured and I(t) = excess rainfall rate. Thus, equation (2) becomes:

Q(t) = A H I(T) MÇtzJO d T (9) 0 dt

and equations (7) and (8) reduce to the simple result:

9S(t) / dt = 1 (10) 0 dt

S(°°) = 1 (11)

A comparison of equation (9) with equation (18a) of Dooge (1973) shows that 9S(t)/3t is the instantaneous unit hydrograph (IUH) of linear hydrological systems theory. Thus, equation (10) states that the area under the IUH is unity, and equation (11) states that the S-curve has an asymptote of unity as t •> °°.

Nash (1958) uses classical unit hydrograph theory to obtain equation (9). On the other hand, Dooge implies that equation (9) and unit hydrograph methods are equivalent since they are both based upon the same two assumptions, viz. that Q and I are related in a linear way and that the hydrological system is time-invariant. These assumptions are mathematically equivalent to requiring that Q and I appear linearly in a governing differential equation whose coeffi­cients are independent of time. An alternative proof of this equivalence can be obtained by showing that an exact evaluation of the integral on the right side of equation (9) leads to a set of equations that are identical with the equations used in unit hydro-graph applications. Thus, if I(t) is given by:

I(t) = I for 0 < t < T

= 0 for T < t < °° (12)

in which In = constant, excess rainfall intensity and T = duration of excess rainfall, then equations (3), (9) and (12) give the exact

334 Bruce Hunt

result :

Q(t)/AI = S(t) for 0 sç t £ T

= S(t) - S(t - T) for T g t £ °° (13)

Equation (13) can be used either to compute Q(t) when the S-curve is known or to compute S(t) when Q(t) is known. Since the calcula­tion of Q(t) from a known S-curve is relatively straight-forward, we will be concerned mainly with the more difficult inverse problem in which the S-curve is calculated from a known hydrograph, Q(t),

If the constant time step is denoted by At, and if the time step equals the duration of excess rainfall (At = T), then equation (13) shows that the S-curve coordinates can be calculated with the recursion formulae:

SQ = 0 (14)

Q S = -v- + S , for n = 1,2,3, . ... (15) n AI n-1

in which a subscript n indicates that a variable is calculated at the time t = nAt. Repeated applications of equations (14) and (15) show that these equations are equivalent to the result:

Q, + Q9 + Qo + - - • •• • + Q

s = _t _£ £ _ J 1 for n = 1,2,3, ..... (16) n AI„

Equation (16) gives, apart from a multiplicative scaling factor, the algorithm that is commonly used to compute S-curve ordinat.es from a known hydrograph,. In practice, though, it is more efficient to use equation (15) since that equation requires only one addition for each value of Sn that is computed,, A similar algorithm, without a formal statement of equations (14) and (15), is used by Linsley et al. (1975)

If the excess rainfall duration is twice the time step (T = 2At), then equation (13) gives the following formulae for S :

SQ = 0 (17)

si = V A Io <18)

Q

s = _S_ + s for n = 2,3,. . (19) n AI n-2

Again, S-curve coordinates are computed most, efficiently by using equations (17) to (19), but repeated application of these formulae leads to the alternative result:

The meaning of oscillations in S curves 335

Q, + Q„ + Q , + - - - + Q 1 3 5 n T O E

S = —— • f o r n = 1 , 3 , 5 A I 0

AIo f o r n = 2 , 4 , 6 , . . . ( 2 0 )

Equation (20) gives the usual algorithm for computing S-curve ordinates.

Equation (16) shows that an error in one value of Qn in this equation will be neither damped nor amplified in later calculations but will simply reappear in each subsequent value of Sn. On the other hand, an error in one value of Qn in equation (20) will reappear only in every second value of Sn. Since this error will be neither damped nor amplified as time increases, it can be concluded that one error in Qn in equation (20) will lead to undamped oscilla­tions in the S-curve ordinates. In general, equation (13) can be used to show that an error in Q n will create undamped oscillations with a period of T whenever the duration of excess rainfall exceeds the time step (T > At) .

Equations (8) and (11) imply that undamped oscillations will not occur in an S-curve calculated for a linear, time-invariant system. Thus, the error referred to in the previous paragraph is, in general, any difference between measured values of Qn and values of Qn that would result from solving a linear differential equation with time-independent coefficients. If T > At, then twice the amplitude of the resulting oscillations will give an estimate for the error magnitude in the dimensionless hydrograph. On the other hand, if T = At, then oscillations will not appear in the computed values of Sn and the error magnitude cannot be estimated. These conclusions remain unchanged if an error is made in specifying either IQ or T in equation (12) since the result can always be interpreted as an error in Q(t) for the specified function l(t). It should also be pointed out that decreasing the time step, At, will increase only the resolution but will not affect the solution accuracy since equation (13) gives the exact result of using equation (12) in equation (9). Finally, it is worth noting that oscillations in the S-curve will lead to larger oscillations in the IUH since the IUH is calculated from the slope of the S-curve. Thus, the well-known difficulty of pronounced oscillations appearing in an IUH as the number of terms in an approximating series expansion is increased may sometimes be the logical result of obtaining a more exact approximation to the IUH when errors are present in Q(t). Evidence for this last state­ment can also be found in a study by Blank et al. (1971), where the same pattern of oscillations is found in an IUH when using each of two different methods to calculate the IUH.

A more general form of equation (13) can be obtained from equation (9) by assuming that I(t) is a discontinuous, piece-wise constant function upon the interval 0 £ t < °°. In this case, the following exact result is obtained:

QQ/A = 0 (21)

336 Bruce Hunt

Q /A = S AIQ (22)

Q /A = Sxr AI + £N~* SM AI for N = 2,3,... (23) N N O n=l N-n n

in which Al n = step increase in I at the time t = nAt (Aln< 0 for a step decrease in I). These equations are useful for the machine calculation of Q(t) from a known S-curve. However, the writer has not found these equations useful for the inverse problem of calcula­ting S(t) from specified values of Q(t) and I(t). This is apparently because curves for Q(t) that result from functions more complicated than equation (12) are apt to contain numerous errors that add and subtract in a complicated way when equations (21) to (23) are inverted to calculate S .

n

NUMERICAL EXAMPLES

It is well known among hydrologists that oscillations often occur in the calculation of S-curves. Linsley et al.(1975) suggest that these oscillations are the result of using an incorrect excess rainfall duration. Since an S-curve is shown by equation (4) to be the outflow hydrograph for a storm of constant intensity over the time period 0 £ t < °°, and since this hydrograph is obtained in practice by superimposing (a large number of times) the hydrograph from a storm of finite duration, then using an excess rainfall duration that is too long or too short will give a storm hyetograph with periodically spaced spikes or gaps, respectively. This gives a physically appealing explanation for the occurrence of oscillations in S-curves. However, it was shown in the previous section that oscillations can also result if the measured hydrograph used to generate the S-curve is not a hydrograph that can be generated by a linear differential equation with time-independent coefficients. Numerical examples will be used in this section to illustrate this statement.

An obvious way to illustrate the effect of hydrograph errors upon S-curve calculations is to calculate an S-curve and downstream hydrograph from a linear differential equation that has time-indepen­dent coefficients. Then it can be seen that exact ordinates for S(t) are obtained from equation (13) by using the calculated solution for Q(t), and a perturbed value of Q(t) can be used to show the effect of an error in Q(t) upon the S-curve, Since accelerations are almost always negligible in flood routing problems, an appro­priate differential equation is given by the diffusion approxima­tion. Weinmann & Laurenson (1979) give the resulting equation:

n 32Q _ 9Q 3Q D — - c — + —- - cq (24)

o, 2 ox dt dx

The new variables introduced by equation (24) are D = diffusion coefficient, c = kinematic-wave velocity, q = lateral inflow per unit length of the river and x = distance downstream from the river origin. The downstream flow rate is Q(x,t), and it will be assumed

The meaning of oscillations in S curves 337

that D and c are constants. Appropriate initial and boundary conditions for this problem are:

Q(x,0) = 0 for 0 < x < °° (25)

Q(0,t) = 0 for 0 < t < » (26)

Thus, as in the unit hydrograph method, Q(x,t) is the difference between the actual flow rate and the flow that would have been measured in the river if no storm had occurred.

The initial-valued boundary-value problem given by equations (24) to (26) is completed by specifying q. Since an S-curve results from an excess rainfall with an infinite duration, q will be specified as:

q = AIQ/L for 0 < t < o° (27)

in which L = value of x for which a downstream hydrograph is being computed, A = area of the basin that contributes runoff to the hydrograph and Io = constant excess rainfall intensity. Thus, equation (9) shows that the S-curve is given by:

S(t) = Q(L,t)

AI for 0 < t <

0 (28)

The solution of equations (24) to (27) is given by:

Q(x,t) x, D ct + x AI L 6 { 2L

erfc ct + x

2/Dt

ct - x 2L

erfc ct

2/Dt

(29)

in which erfc = complementary error function. Equation (29) is readily seen to satisfy equations (25) and (26) , and a little more effort shows that equations (24) and (27) are also satisfied. Thus, equations (28) and (29) give the following solution for the S-curve:

S(t) = 1 - e

cL D ct + L

2L

f erfc

ct

2/Dt

ct - L 2L

erfc ct - L

2/Dt

(30)

A dimensionless plot of equation (30) is shown in Fig.l. One notable feature of this plot is that the S-curve plotted from equa­tion (30) does not have a small region of positive curvature near the origin that is usually observed in S-curves derived from field-measured hydrographs. The writer is unable to make any further comment upon this detail except to suggest that the missing region of positive curvature may be the result of assuming that q in equation (24) is uniformly distributed in space or it may be the result of nonlinearities or time-dependent coefficients in the governing differential equation. For example, Henderson & Wooding

338 Bruce Hunt

1.0

0.8

Sit)

0.6

O.i

0.2

0 0 0.2 O.i 0.6 0.8 1.0 1.2 1.L 1.6 1.8 2.0 2.2

ct/L

Fig. 1 A dimensionless plot of S-curves calculated from equation (30).

(1964) obtain hydrographs from nonlinear kinematic-wave solutions that have positive curvature near the origin, and introducing the inflow, q, in equation (24) at a single point upstream will also give solutions with positive curvature near the origin.

A dimensionless plot of excess rainfall rate for a storm of finite duration is shown in Fig.2, and the S-curve shown in Fig.l for

1

lit) h

0 0.2 O.i 0.6

ct/L

Fig, 2 A dimensionless plot of excess rainfall rate.

D/(cL) =0.1, the excess rainfall rate shown in Fig.2 and equation (13) give the dimensionless hydrograph shown with a solid curve in Fig.3. Thus, Figs 1 and 3 give an exact solution for a linear differential equation with constant coefficients for the excess rainfall rate shown in Fig.2. In other words, Figs 1-3 provide a solution that is suitable for testing the effect of errors in generating an S-curve from equation (13). Numerical values for the S-curve and downstream hydrograph are also given in Table 1.

It is easily verified that the third column in Table 1 and equation (13) can be used to recalculate the S-curve ordinates in the second column of Table 1. This is true regardless of whether At = T or At = T/2, which is a result that is not surprising since equation (13) and the second column in Table 1 were used to calculate the third column in the first place.

cT^°> 0.1-

The meaning of oscillations in S curves 339

0.2 O.i 0.6 0. 2.0 2.2 1.0 1.2 1.1 1.6 1.1 ct/L

Fig. 3 A dimensionless hydrograph computed f rom Figs 1 and 2 for D/(cL) = 0 .1 .

Table 1

c t /L S i t ) * Q ( t ) / A I 0 t

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

0 0.200 0.399 0.582 0.727 0.829 0.896 0.938 0.963 0.979 0.987 0.993 0.996

0 0.200 0.399 0.382 0.328 0.247 0.169 0.109 0.067 0.041 0.024 0.014 0.009

* Values computed for D/{cL) = 0.1. t Values computed for I (t) shown in Fig. 2.

A perturbation of 0.05 at ct/L = 0.2 in the downstream hydrograph

gives the hydrograph shown with a dashed line in Fig.3. The result

of using this perturbed hydrograph in equation (13) to calculate the

S-curve is shown in Fig.4. As predicted by equation (20), every

Perturbed Solution

S(tl

0.2 O.i. 0.6 0.8 1.0 1.2 1.1, 1.6 1.8 2.0 2.2 ct/L

Fig. 4 S-curve solution using the perturbed hydrograph shown in Fig. 3.

340 Bruce Hunt

second S-curve ordinate differs from the exact S-curve ordinate by an amount equal to the error in Q/(AIQ) at ct/L =0.2 in Fig.3. Thus, the period of oscillation exactly equals the duration of rainfall excess in Fig.2 (cT/L = 0.4), and twice the amplitude of the oscillation gives the exact value of the error in Q/ ( A I Q ) . Evidence that the perturbed hydrograph is not the solution of some other linear differential equation with time-independent coefficients is provided by the fact that the undamped S-curve oscillations plotted in Fig.4 give a behaviour at t = œ that is not consistent with equation (11).

The result of using an incorrect excess rainfall duration, T, can be illustrated by using the hyetograph plotted in Fig.5. Since the runoff duration in Fig.5 is 3/2 times the runoff duration in Fig.2, the ordinates in Fig.5 are 2/3 times the ordinates in Fig.2 in order to ensure that the runoff volume and S-curve asymptote remain unchanged. The result of using Fig.5 and equation (13) to calculate the S-curve is shown in Fig.6. As noted earlier, this problem is

Fig. 5 An incorrect excess rainfall rate d is t r ibut ion.

s ft)

0 0.2 O.i 0.6 2.0 2.2 1.0 1.2 1.1, 1.6 1.8 ct/L

Fig. 6 S-curve solution using the excess rainfall rates shown in Figs 2 and 5 and the hydrograph in Fig. 3.

similar to the previous problem since the hydrograph can be looked upon as an incorrect hydrograph for the given excess rainfall rate in Fig.5. Thus, the oscillations are observed to have a period equal

The meaning of oscillations in S curves 341

to the excess rainfall duration, and the error in the dimensionless hydrograph can be expected to be about twice the amplitude of the oscillations in the S-curve. This last statement is verified in Fig.7, in which exact hydrographs are plotted for the S-curve in Fig.l [D/(cL) = 0.1] and the excess rainfall rates in Figs 2 and 5, The maximum difference between the two hydrographs is Q/(AIQ) = 0.133 at ct/L = 0.4, which compares with a value of about 0.2 for twice the maximum oscillation amplitude in Fig.6. In general, of course, twice the amplitude of the oscillations only gives an order of magnitude estimate for the maximum hydrograph error since errors occurring at a number of different times can be expected to interact in a more complicated manner than would happen if only one isolated error occurs. Nevertheless, the presence or absence of oscillations and the magnitude of oscillation amplitudes in an S-curve can be expected to give a qualitative indication as to the suitability of unit hydrograph theory for predicting flood flows in any given river.

0 0.2 O.i 0.6 0.8 1.0 1.2 1.i 1.6 1.8 2.0 2.2 ct/L

Fig. 7 Hydrographs computed f rom the S-curve in Fig. 1 and the excess rainfal rates in Figs 2 and 5.

CONCLUSIONS

The Duhamel superposition integral has been used to derive some exact solutions in unit hydrograph theory. These solutions and numerical examples have been used to show that oscillations can be expected to occur in a derived S-curve whenever the time step is less than the excess rainfall duration (At < T) if the hydrograph is not the exact solution of a linear differential equation with time-independent coefficients. The S-curve oscillations will have a period equal to the excess rainfall duration, T, and an amplitude that is of the order of half of the maximum difference between the measured dimensionless hydrograph and the hydrograph that would have resulted from the solution of a linear differential equation with time-independent coefficients. Thus, S-curve oscillations give a qualitative indication of the suitability of unit hydrograph theory for predicting flood flows in any given river. Since an IUH is calculated from the slope of an S-curve, oscillations in an S-curve will lead to even larger oscillations in an IUH. This suggests that great difficulties can be encountered when trying to calculate

342 Bruce Hunt

an IUH from experimental data. On the other hand, neither IUH's nor finite-period unit hydrographs are necessary for calculating flood flows since S-curves can be calculated directly from a measured hydrograph and excess rainfall duration by using equation (13). Once calculated, the S-curve ordinates and equations (21) to (23) can be used to calculate the hydrograph for any complex storm. This explains in part why oscillations in an IUH have a relatively small effect upon a calculated runoff hydrograph. Since the calculated hydrograph depends only upon the excess rainfall rate and S-curve ordinates, and since S-curve ordinates are given by the integral of the IUH, large oscillations in an IUH will give considerably smaller oscillations in both the S-curve and calculated hydrograph.

ACKNOWLEDGEMENT The writer would like to thank Dr R.H.Spigel for making a number of helpful comments and suggestions.

REFERENCES

Blank, D., Delleur, J.W. & Giorgini, A. (1971) Oscillatory kernal functions in linear hydrologie models. Wat. Resour. Res. 7 (5), 1102-1117.

Dooge, J.C.I. (1973) Linear Theory of Hydrologlcal Systems, p.24, 84-89. US Dept of Agriculture, Tech. Bull. no. 1468, Washington, DC.

Hall, M.J. (1977) On the smoothing of oscillations in finite-period unit hydrographs derived by the harmonic method. Hydrol. Sci. Bull. 22 (6), 313-323.

Henderson, F.M. & Wooding, R.A. (1964) Overland flow and ground­water flow from a steady rainfall of finite duration. J. Geophys. Res. 69 (8), 1531-1540.

Hildebrand, F.B. (1962) Advanced Calculus for Applications,451-453. Prentice-Hall, Inc., Englewood Cliffs.

Linsley, R.K., Kohler, M.A., & Paulhus, J.L.H. (1975) Hydrology for Engineers, 2nd edition, 243-245. McGraw-Hill, New York.

Nash, J.E. (1958) Determining run-off from rainfall. Proc. Instn. Civ. Engrs. 10, 163-184.

Sherman, L.K. (1932a) Steam flow from rainfall by the unit-graph method. Engng News Rec. 108, 501-505.

Sherman, L.K. (1932b) The relation of hydrographs of runoff to size and character of drainage basins. AGU Trans. 13, 332-339.

Weinmann, P.E. & Laurenson, E.M. (1979) Approximate flood routing methods: a review. J. Hydraul. Div. ASCE 105 (HY12), 1521-1536.

Received 4 April 1984; accepted 27 February 1985.