TYPE OF MATHEMATICAL MODELS
16.1 TYPE OF MATHEMATICAL MODELS
The mathematical models in Hydrology can be classified as: (a) Stochastic models (b) Deterministic models In the stochastic model, the chance of occurrence of the variable is considered thus
introducing the concept of probability. A stochastic model is time-dependent while the probabilistic model is time-independent. For the time-independent probabilistic process, the sequence of occurrence of the variates involved in the process, is ignored and the chance of their occurrence is assumed to follow a definite probability distribution in which the variables are considered pure random. For the time-dependent stochastic process, the sequence of oc- currence of the variates is observed and the variables may be either, pure random or non-pure random, but the probability distribution of the variables may or may not vary with time. For example, the flow duration curve procedure is probabilistic, whereas the flood routing through
a reservoir is a stochastic procedure. In the deterministic models, the chance of occurrence of the variables involved is ig-
nored and the model is considered to follow a definite law of certainty but not any law of probability. For example, the mathematical formulation of the unit-hydrograph theory is a deterministic model.
Both the stochastic and deterministic models can be sub-classified as (i) Conceptual models
(ii) Empirical models
In conceptual models, a mathematical function is conceived based on the consideration of the physical process, which when subjected to input variables, produces the output vari- ables. For example, a conceptual catchment model of rainfall-runoff relationship can be de- scribed by
...(16.1) where i(t) = input, i.e., rainfall Q (t) = output, i.e., runoff φ = system operator, i.e., it represents the operation performed by the system to yield the output for the given input. Empirical models are based on empirical relationships or formulae like Dickens, Ryves,
φ[i(t)] = Q(t)
Inglis, etc., where one composite coefficient takes into account all the variables affecting
MATHEMATICAL MODELS IN HYDROLOGY
possible flood peaks in a catchment; all the complex physical laws involved may not be considered here.
A conceptual or empirical model could be either de- terministic or stochastic or a combination of these. For example, the model for predicting sedi- ment transport is a combination of deterministic and stochastic components, which are often based on empirical relationships, where as models based on regression analysis are stochastic conceptual models with a strong deterministic approach. Models concerning the propagation of flood wave and Nash model (cascade of linear reservoirs) are the examples of deterministic conceptual models, while models based on synthetic unit hydrograph are the examples of de- terministic empirical models.
Examples of mathematical models.
Optimisation of model and efficiency of model . A system is a set of elements organised to perform a set of designated function in order to achieve desired results. The project formula- tion of the system may be called system design. The objective of system design is to select the
combination of system units or variables that maximises net benefits in accordance with the requirement of the design criteria. The design so achieved is known as the optimal design. The optimisation is subject to the requirements of the design criteria or constraints that are im- posed. The constraints may be technical, budgetary, social or political and the benefits may be real or implied. When an objective is translated into a design criterion, it may be written in the form of a mathematical expression known as the objective function.
For mathematical model, the objective function which should be optimised is given by the equation
F = ∑ (Q′ ) 2 i –Q i
where Q′ i = actual observed value Q i = value predicted by model The objective of the optimisation procedure is to obtain such sets of parameter values,
which reduce the least squares objective function F to zero. For testing the significance and reliability of the optimised parameter values, statistical tests, predictive tests and sensitivity
tests are often employed. Nash, et al (1970) have suggested the use of model efficiency (R 2 ) to evaluate the performance of the model.
F 0 2 = ∑ (Q i – Q )
where
...(16.3a)
= mean of n observed values
R 2 is analogous to the coefficient of variation and is proportional to initial variation accounted by the model. The efficiency of a separable model part (r 2 ) can be judged by a change in the value of R 2 resulting by the insertion of that part or by the proportion of residual vari- ance accounted for by its insertion.
r 2 = 2 1 = 1 2 2 ...(16.4)
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Predictive testing on a split sample basis is adopted in which the available data is di- vided into two parts; the first part is used to estimate the optimised parameter values and then, these are used to predict or reconstruct the second part of the data record. Sensitive tests generally test the effect of varying the optimised parameter value or the data on the objective function. This gives an idea of stability of the optimised parameter values or data error effects. Two studies on these lines are given below.
Excess rainfall-direct-runoff model –A model study was done at Roorkee (1976) using the data of a small catchment. Fig. 16.1, having a non-uniform-rainfall distribution. The val- ues of n and K obtained by the method of moments in a Nash Model (1957)* were not able to
Sub area A 2
To Badnpur
Karpa Karpa
Kherwari Kherwari
Sub area A
Betul Wagh Wagh
oli oli
Catchment
Bar kher
boundary
Gh at Baroli
Bridge
Sub area A
no. 566
3 To Nagpur
Rain gauge stn.
To katol
a. Small catchment divided into three sub areas - A , A & A 1 2 3
Sub area A 2 Sub area A 1 I 2 I 1
Linear reservoirs
n = 5.32 2 n = 5.22 1 k = 0.45 hr 2 k = 0.435 1 Sub area A 3
hr
o 2 o 1 n = 4.81 3
Linear
k = 0.279hr
channel
O 3 O+O+O 1 2 3
b. Model of linear reservoirs and linear channel Fig. 16.1 Rainfall-runoff model (Roorkee, 1976)
simulate the peak of the observed hydrograph, though the values of the model efficiencies R 2 = 83.7% and 82.3% were obtained for the hydrograph used to derive the values of n and K, and the hydrograph of direct runoff, respectively. Then the catchment was divided into three sub- areas and each sub-area was modelled by means of cascade of linear reservoirs of equal stor- age coefficient. The values of n and K for each sub-area were obtained by using physiographic
* Nash (1957) used a series of n linear reservoirs of equal storage coefficient K, to obtain the instan- taneous unit hydrograph assuming a lumped input and linear, time-invariant, deterministic system.
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features of the catchment. The outflow from sub-areas A 1 and A 2 was combined together and then led through a linear channel having a translation constant T; it was then combined with outflow of sub-area A 3 to obtain the values of direct runoff for the whole catchment. The model efficiency R 2 = 82.3% and 90% for the first and second hydrographs, respectively. However, further analysis had to be made for investing the effect of loss assumptions and non-uniform rainfall distribution, and possibly inclusion of translation concept by dividing the sub-areas by means of isochrones and representing them by combination of linear channel and linear reser- voir in series.
Rainfall-runoff model based on retention concept . The retention concept of dividing the rainfall into runoff and non-runoff volumes is based on the physical processes of soil moisture retention. Model based on the retention concept was used by Seth (1972) for simulating the rainfall-runoff process of a small catchment. Its main components were those used to represent the retention capacity Y and evapotranspiration capacity E. Retention storage S was defined
as that volume of water present in a natural watershed, which is not runoff storage. The maximum storage volume V of the retention storage of the catchment was subdivided into V 1
and V 2 for the upper and lower layer of soil, respectively. The moisture storage at any time, i.e., retention storage in the upper and lower layers were S 1 and S 2 , respectively, and the ratio of S 1 /V 1 and S 2 /V 2 characterised the moisture status of the upper and lower layers, respectively
V =V 1 +V 2 ...(16.5) S =S 1 +S 2 ...(16.5a)
Precipitation
Rainfall Rainfall
Other abstractions Other abstractions excess (P excess (P net net ) )
Infiltration (F) Infiltration (F)
Subsurface Subsurface
Deep Deep
flow flow
percolation percolation Ground water Ground water
Prompt Prompt
Delayed Delayed
interflow interflow
interflow interflow
Direct surface Direct surface
Base flow (BF) Base flow (BF)
runoff, DSR(Q ) runoff, DSR(Q ) d d
Total runoff Total runoff (TR or Q) (TR or Q)
Fig. 16.2 Rainfall-runoff system
These components together with other components of the model gave a good perform- ance when about a year’s data of 2560 data pieces of 3 hourly values of rainfall, runoff and potential evaporation of Grendon Underwood Catchment of Institute of Hydrology (UK), were
used. The values of F = 14.07 mm 2 and R 2 = 0.912 obtained after optimisation run, compared quite favourably with those obtained by using the Stanford Model for the same data. The model values of soil moisture retained in soil gave almost a similar pattern as the observed soil
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moisture. However, to take into account the full range of seasonal variations, it would be desirable to include a wide range of antecedent and storm period conditions existing perhaps over several years.
Rainfall-runoff system model . The rainfall to runoff transformation as a system is shown in Fig. 16.2.
A famous computer model is the Stanford Watershed Model (Crawford and Linsley, 1966). This model is refined progressively and is now a very comprehensive model based on water budgeting. By using hourly precipitation data and daily evapotranspiration as the main inputs, the model is programmed to produce hourly streamflow. Soil, vegetation, land use, etc., are all accounted for by a set of parameters, the values for which are progressively optimised by search technique. The flow diagram for the model is shown in Fig. 16.3 a which is, typical flow chart for a rainfall-runoff model of the explicit soil moisture accounting type, ESMA, redrawn from Linsley and Crawford (1974).
KEY
Precipitation potenial
Functions Evapo-
Snow melt
Channel ration
inflow Evapo-
Lower zone or
transpi-
zone
ground water
zone depletion
Active or deep ground water storage
Evapo- transpi-
Channel Channel
storage
inflow outing
Simulated inactive ground
Deep or
streamflow
water storage
Fig. 16.3 Flow diagram of stanford watershed Model IV, ESMA (after Linsley and Crawford, 1974)
MATHEMATICAL MODELS IN HYDROLOGY