6-мавзу. Математик моделлаштириш элемент­лари

c) According the way of considering the time, the following models stand out :

1. 
   discrete models;
   
2. 
   continuous models.
   

It has been proved that the systems have a dynamic character, the input, output and the system's states being functions of the time t. Obviously, the variable t is continuous belonging to the real numbers, and thus the real systems are continuous versus time. The use of the continuous models raises however special difficulties; that is why in practice, discrete models are used frequently, the variable t being discretized. This approximation of continuous processes through discrete models allows the use of numeric solving methods.
In hydrology and water management, as well as in the case of other technical systems, the discretization of the time's axis in intervals, whose value depends on the analysed process, is frequent. For example, in the case of high floods, this interval takes values from one to several hours, in function of the river basin area; for the allocation of water resources from a reservoir, the time step is usually one month. For regional aquifers, the discretization step may even reach 2-3 months.
d) According to the degree of knowledge of the analysed systems, the following models are used:

1. 
   physically based models (white-box models);
   
2. 
   engineering models of various types:
   
   1. 
      input-output models (black-box);
      
   2. 
      input-states-output elementary models;
      
   3. 
      conceptual models (grey-box).
      

The physically based models describe rigorously a process or phenomenon through some differential equations or equations with partial derivates; the solutions are obtained by analytical or numeric integration. The physical processes of the runoff formation on slopes or the processes of water's infiltration into the soil may be studied with such models. In fact, these models are input-states-output physically based models. They reflect quite exactly the analysed phenomenon or process, being characterized by complexity and implicitly by difficult solving; for the calibration of the parameters of these models, experimental measurements are necessary.
Engineering models have as a purpose to obtain the output O(t) in function of the input I(t), without a detailed study of the system's inner processes.
The function that transforms input into output within the framework of the black-box models type is called kernel function or weighting function. The relations being used have different mathematical expressions, as the variables may be discrete or continuous.
For continuous variables, the output O(t) is calculated using the convolution integral, called also the Duhamel integral:

(7.1)

where U(t) represents the weighting function.
This integral has the following symmetry property:

(7.2)

meaning that:

(7.3)

In the case of discrete variables, the convolution integral becomes:

(7.4)

where U_i are the discretized values of the weighting function.
The most widespread weighting function in hydrology is the unit hydrograph, used in the modelling of hydrological processes.
If within the input and the output functions are replaced with their Laplace transformations, the notion of transfer function is used. In this case, the relation between input and output is written as it follows:

(7.5)

where:
	ai	parameters attached to the output values;
	s	a complex variable;
	O*(s)	the output value;
	bi	parameters characterizing the model's input;
	I*(s)	the input value.

The ratio between the Laplace transformations of the system's output and input:

=

(7.6)

represents, by definition, the transfer function of the system and it allows the transformation of the input I*(s) into the output O*(s):

=

(7.7)

We remind here that the Laplace transformation of a function f(t) is the function Φ(s), defined by the integral:

(7.8)

The denomination of transfer function may be used even in the cases when the input and output variables are expressed as real numbers; even the runoff coefficient, which is the ratio between the effective precipitation and the total precipitation represents a transfer function. Any operator, which if applied to the input transforms it into output, may be considered as a transfer function.
As input-states-output elementary models in hydrology the most common are those models for which the runoff coefficient depends on the evolution of the soil's humidity, which characterizes the system's state.
Conceptual models consist in the detailing of the processes that take place within the system, after its decomposition into elements. The evolution from input-output or input-states-output elementary models to conceptual models is actually the equivalent of the opening (or the lighting) of the black-box. The system's components are linked in series or in parallel; each component transforms its input in specific output. Conceptual models move the use of kernel functions or transfer functions from system level to subsystems level, which allows them to be considered a generalization of the previous models.
The name of conceptual models refers to the way in which the specific transformation mechanisms of every component, as well as the links between components, are imagined or conceived.
Typical examples of conceptual models in hydrology are the reservoir type models. The water coming from precipitations goes successively through a series of interconnected reservoirs (vegetal layer, snow reservoir, soil layer, aquifer, hydrographical network), each reservoir being emptied according to its own laws.