A NEW APPROACH TO NON-LINEAR PROBLEMS IN SURFACE WATER HYDROLOGY
"Hydrologie Systems with Uniform Non-linearity"
James C. I. D O O G E Ireland
SUMMARY
A new approach based on the concept of uniform non-linearity is proposed for the analysis and synthesis of non-linear hydrologie systems. Preliminary analysis of some laboratory and field data already published indicates that the assumption of uniform non-linearity is reasonable in these cases.
RÉSUMÉ
Une approche nouvelle fondée a le concept de « uniform non-linearity » est proposée pour l'analyse et synthèse des systèmes non-linéaires hydrologique. L'analyse prélimi- naire de quelques essais dans le laboratoire et dans les bassins, qui ont été publiés, démontrent que l'assumption de « uniform non-linearity » est raisonable dans ces examples.
LINEAR METHODS
Several well-developed techniques are available for the analysis of linear time- invariant systems. In such cases, the response of the system to any input can be obtained by the convolution of the input with the impulse response of the system. Thus, the behaviour of a linear time-invariant system is completely determined by the impulse response. This principle is implicit in the unit hydrograph approach, first used by Sherman in 1932 C1). Over the years, unit hydrograph techniques have been refined and the subjective elements gradually removed so that there exists today a well-developed theory of linear hydrologie systems. The behaviour of any linear hydrologie system can be completely characterized by the instantaneous unit hydrograph and the form of the latter determined by means either of methods based on orthogonal functions or of methods based on least squares (2).
The development of objective methods of linear analysis has been paralleled by the development of the synthesis of hydrological systems. One method of synthesis is that of mathematical simulation. In the latter method, an attempt is made to simulate the action of the watershed by a model consisting of a simple arrangement of simple linear elements. Most of the linear models used to simulate surface runoff consists of some simple combination of linear reservoirs, (i.e., elements in which the outflow is propor- tional to the first power of the amount stored) and linear channels (3). It can be shown (4) that the impulse response of any damped linear system can be represented as the sum of a series of Laguerre function terms and hence, as the sum of a series of gamma distri- bution terms. This means that any damped linear system can be simulated by a model consisting entirely of some arrangement of linear storage elements. It has been found in practice that a single gamma distribution term gives a good representation of the impulse response, both of catchments (5) and of channels (6).
NON-LINEAR METHODS
The fact that the basic dynamic equation of open channel flow is highly non-linear has always cast doubts on the validity of unit hydrograph methods as a basis of pre- dicting the outflow from a watershed. Little detailed quantitative evidence is available
on the linearity or non-linearity of the process of watershed runoff but the indications are that the action is highly non-linear at small flows but almost linear for higher flows C»8). Apart from the non-linear terms in the dynamic equation of open channel flow, the overall behaviour of catchments is likely to be non-linear due to the presence of thresholds in the system.
If the assumption of linearity is abandoned, one loses all the advantages of linear mathematics and the techniques based on this mathematics. General non-linear methods of analysis and synthesis have been described (9>10) but these have not up to the present been carried to the point of application in applied hydrology. These general non-linear methods are extremely complex in comparison with the corresponding linear methods.
Some writers such as Kulandaiswamy (u) have tackled the question by the method of linearisation. In this approach, the system is analyzed by linear methods and then the parameters of the unit hydrograph are correlated with the average intensity of the input.
In the present paper, a new approach is outlined. In this new approach a special class of non-linear systems is examined so as to avoid much of the complexity of a gener- al non-linear approach but with the hope of accounting for some of the non-linear effects which are of importance in applied hydrology.
T H E CONCEPT OF UNIFORM NON-LINEARITY
A uniformly non-linear time-invariant system may be denned as one whose actions can be simulated with sufficient accuracy by a model consisting of some arrangement of equal non-linear storage elements. The outflow from each of these storage elements would be related to the storage in the element by the equation
q=aS
c(1)
Since such a system is non-linear, the properties of proportionality and superposition will not hold. It can, however, be shown that for a uniformly non-linear system the property of proportionality will hold provided that the time scale has been previously transformed in accordance with the intensity of input. If the average intensity of the input is given by ro, then we can define a characteristic storage as
and a characteristic time by
^o So
It should be noted that once the characteristic inflow of ro has been chosen, the charac- teristic storage and the characteristic time can be calculated by using only the parameters a and c of the non-linear storage elements, which are the basis of the simulation.
The next concept to be introduced is that of similar inputs. Two inputs may be said to be similar, in the sense of uniform non-linearity, if they have the same dimensionless input when the ratio of input to average input rate is plotted against the ratio of time to characteristic time. It can be shown that if we have a system with uniform non- linearity, then the dimensionless outputs corresponding to two similar inputs of different average intensity will be the same.
It is not essential that the average input rate or the characteristic time as defined by equation (3) be used to characterize the dimensionless inputs or dimensionless outputs.
Any definition could be used for the characteristic input rate or the characteristic time
(1)
(3)
provided that the definition is consistently applied in a given case. Thus, the peak input or any other objectively defined input could be used as the charateristic input rate; and the characteristic time could be taken as time to peak or the lag time, etc.
APPLICATION TO OVERLAND F L O W
The problem of overland flow is one of great importance, both in the analysis of the hydrological cycle and in engineering design. Though it can be simply stated, the solu- tion of the problem is complex. A solution has usually been sought on the basis of some empirical assumption which simplifies the dynamic equation. Two important simplified approaches much quoted in the hydrological literature are the Horton-Izzard approach (12,13) a n (i the kinematic wave approach (14). Though these approaches give quite different results for any given case, it can be shown that each of them involves a model which has the property of uniform non-linearity.
The Horton-Izzard approach assumes a definite relationship between the average detention and the outflow throughout the hydrograph. This is equivalent to treating the overland flow system as a single non-linear reservoir and hence, as a very special case of uniform non-linearity. The kinematic wave approach on the other hand assumes that any point on the plane of overland flow, there is a fixed relationship between local discharge and local depth. This also results in a unique dimensionless outflow when the ratio of outflow to inflow is plotted against the ratio of time to the time of rise. Accorr- ingly, the kinematic wave solution may be classed as a particular case of uniform non- linearity. The fact that in linear flood routing the kinematic wave solution produces a pure translation would suggest that the kinematic solution may be considered as a cas- cade of an infinite number of equal infinitesimal non-linear storage elements.
APPLICATION TO SMALL WATERSHEDS
Some empirical results for small watersheds were examined to see if the results could be plotted in dimensionless forms in terms of characteristic time parameter. If this could be done it would be an indication that such basins could be represented reasona- bly by model systems based on uniform non-linearity.
The empirical results examined were (a) results from a small test basin in the labora- tory reported on by Amorocho and Orlob (9), (b) a digital computer simulation of catchment outflow reported by Machmeier and Larson (15) and (c) unit hydrographs for a number of small watersheds reported by Minshall (16). In all three cases the authors quoted their results as being indicative of a high degree of non-linearity in the hydrological systems being studied. Preliminary results indicate that in all three cases dimensionless plottings show that a uniformly non-linear model would give a very good representation of the results. These preliminary results give every encouragement that it may be possi- ble, by use of the assumption of uniform non-linearity, to acount to a large extent for the non-linear behaviour of hydrologie systems without the full complexity of a general non-linear systems analysis.
REFERENCES
(!) SHERMAN, L.K., Stream Flow from Rainfall by the Unit Graph Method, Engi- neering-News Record, 1932.
(2) O ' D O N N E L , T., Methods of Computation in Hydrograph Analysis and Synthesis, Part II - Recent Trends in Hydrograph Synthesis, Proc. and Info. No. 13, I N O , The Hague, 1966.
(3) DOOGE, J.C.I., A general Theory of the Unit Hydrograph, / . of Geophys. Res., 64(2):241-256, 1959.
(4) DOOGE, J.C.I., Analysis of Linear Systems by Means of Laguerre Functions, SIAM J. on control, 2(3):39.6-408, 1965.
(5) NASH, J.E., Unit Hydrograph Study with particular reference to British Catch- ments, Proc. Inst. Civil Engin., Vol. 17, p . 249, 1960.
(6) KALININ and MILYUKOV, O raschete neustanovivshegosya dvizheniay vody y otkrytykr ruslakh (On the Computation of Unsteady Flow in Open Channels), Meteorologiya i Gidrologiya (USSR), N o . 10, pp. 10-18, 1957.
(7) ISHIHARA, T. and TAKASAO, T., Applicability of Unit Hydrograph Method to Flood Prediction, Proc. Xth IARH Congress, 2(81), London, 1963.
(8) PILGRIM, D . H . , Radio Active Tracing of Storm Runoff on a Small Catchment, II - Discussion of Results, / . of Hydro!., IV(4), p . 38, 1966.
(9) AMOROCHO, J. and ORLOB, G.T., Non-Linear Analysis of Hydrologie Systems, Berkeley Water Resources Center, Contribution, 40, 1961,
(10) JACOBY, S.L.S., A Mathematical Model for Non-Linear Hydrologie Systems, / . Geophys. Res., 71(20), Oct. 1966.
(u) KULANDAISWAMY, V.C. and SUBRAMANIAN, C.B., A Non-Linear Approach to Runoff Studies, Proc. Intl. Hydrol. Symposium, Fort Collins, Colorado, Sept. 1967, Volume 1.
(12) HORTON, R.E., The Interpretation and Application of Runoff Plot Experiments with reference to Soil Erosion Problems, Soil. Sci Soc. Amer. Proc, Vol. 3, p . 340, 1938.
(13) IZZARD, C.B., Hydraulics of Runoff from Developed Surfaces, Proc. Highway Res. Board, Vol. 26, p. 129, 1946.
(14) HENDERSON, F.M. and WOODING, R.A., Overland Flow and Groundwater Flow from a Steady Rainfall of Finite Duration, / . Geophys. Res., 69(8), April 1964.
(15) MACHMEIER, R.E. and LARSON, C.L., The Effect of Runoff Supply Rate and Duration of Hydrographs for a Mathematical Watershed Model, ASCE Hydr.
Dio. Annual Conference, Madison, Wise. 1966.
(16) MINSHALL, N.E., Predicting Storm Runoff from Small Experimental Watersheds, / . of Hydr. Div., ASCE, 86(HY-8), Aug. 1960.
Intervention de J.E. N A S H
Questions:
The principle of this paper would seem to be obtainable merely by dimensional consideration, and the conclusions would seem not to be restricted to the model of a series of equal non linear reservoirs proposed by Professor Dooge.
The conclusion of the paper would seem to be that if one input and the correspon- ding output are available a series of corresponding input and outputs can be obtained by suitable transformation of the given pair.
This is merely an expression of the principle of similarity which states that provided the parameters of the system are also appropriately scaled the input and output may be transformed according to any arbitrary chosen linear scale relationships. In the model proposed two parameters only occur one of which is numerical and the other (a) (in q = aSc) has complicated dimensions expressable in terms of two fundamental dimensions—rate of discharge and time—the input and output are also expressable in these terms.
Therefore if any linear scales tranformation is applied to the input and output and the parameter a, the input and output so obtained will be compatible with the trans- formed system. If in addition the scales transformation of discharge rate and time) are so chosen that the appropriate scale of (a) is unity the transformed input and output will be compatible with the original system.
Similar inputs (which produce similar outputs) are therefore that class of inputs which can be obtained from one another by scale changes such that 'a' is kept constant.
This principle is clearly equally valid for any model which does not contain two or more parameters of different dimensions of discharge and time.
Reply by professor DOOOE
The main point in the author's paper was that for the case of a non-linear system of known uniform non-linearity, data from a single input-output event would define not only this event but the output due to any similar input, i.e. any input which when transformed according to the known degree of uniform non-linearity would give the same dimensionless input as the measured input. The model suggested in the paper was that of some arrangement of equal non-linear storage elements. The relationship be- tween discharge and storage of each of these elements would be given by
q=aS
c(la)
which can also be written as
a
For the system to be uniformly non-linear the value of c must be the same for each ele- ment but variations in the value of a would not affect the uniformity of the non-linearity.
Neither would it affect the uniformity of the non-linearity if instead of pure storage elements the model were constructed from Muskingum type elements in which the storage would be related to both the inflow and the outflow as foliows:
S=-(<Z)
1/C+ 7(0
1/C(2) a b
For practical convenience it would appear preferable to concentrate, initially at least, on the model indicated in the paper which consists of equal non-linear storage elements.
The author agrees with Mr. Nash that the principle of similar inputs is "equally valid for any model which does not contain two or more parameters of different dimensions in discharge and time". Models composed of elements behaving according to equation (1) or equation (2) clearly fulfil this conditions if the value of c remains constant, i.e., if they are uniformly non-linear. Mr. Nash would appear to suggest that the models proposed by the author form a small proportion of the general class of models obeying the criterion given by him. Certain heuristic arguments leave the author to doubt whether any realistic hydrological models satisfying Mr. Nash's criteria could be constructed other than those made up from elements obeying equation (1) or the more general equation (2). Equations (1) and (2) are certainly sufficient conditions to to guarantee the property under discussion and equation (2) may be a necessary condition. But even if it were not, there would seem to be little advantage in using more complex models at the present time.
