Keywords: Optimization, Dimension Remote sensing, Theorem, Optical channel

Abstract
Questions related with optimization of systems of active remote sensing (lidars, radars, sonars etc.) taking into account of energetic losses of sensing signal in the investigated medium are considered in first part of this submission. Physical model of such systems, envisaging energetic losses is considered.

It is shown, that mathematical model of remote sensing process, according which signal, reflected from the n - th border doesn't contain influence of previous reflections from n-1 borders, could be obtained from considered physical model. In this case, signal reflected from n - th border will contain of influence of systematic energetic losses in homogenous layers of the investigated object.

Theorems, optimizing grade regimes of system's output signals when correction of fading is carried out and is not carried out are proved.

In second part of this submission grounding of synthesis of optimal passive remote sensing systems is given. As a result of held research it was shown, that optimum value of band width of radiometers is existed within transparent wavelength band 3 - 5 mcm.

1. Optimization Of Active Remote, Sensing Systems

1.1. Principle of the measuring systems dimension lowering
Criterion of effectiveness of any class of measuring systems, could be expressed by functional

U = F (X, Y)
where X=(x₁, x₂, . . ., x_n) - vector, characterizing system's parameters, which could be managed; Y=(y₁, y₂, . . ., y_n) - vector, composed of non - managed parameters of a system.

If according to the principle of measuring system's dimension lowering [1], we assume, that vector X=(x₁, x₂, . . ., x_n) characterizes determined class of n - dimensioned systems, so any specific lowering of X's dimension will characterize some subclass of such systems. For lowering of X's dimension we suppose, that some components of vector X are not independent and managed by other independent components of vector X . Thus, vector X is substituted by two vectors: X₁ - independent or managing one, and X₂ - dependent, or managed one. The criterion of effectiveness for such systems is functional of following type

U₁=F(X₁, X₂, Y)
In common case, designating above dependence as
X₂ = ø(X₁) we obtain
U₁F[X₁,ø(X₁), Y]       (1)

Therefore, the task of synthesis is lead to the searching of maximum of functional (1) for subclass of systems.

It should be noted, that two variants synthesis of subclasses are possible: 1. Function ø(X₁) is unknown and should be determined [1]; 2. function ø(X₁) is known [2].

In this submission we consider the second variant of synthesis applied for active and passive systems of remote sensing.

1.2. Model of researched object
A multilayered model, consisting of n - number of various homogenous layers is accepted as a basic model of studied object of active remote sensing. We assume, that sensing signal loses its own energy as a result of two processes:

1. Reflections from mutual border of any two various contiguous homogenous layers.
2. Fading in homogenous layers of studied object.
Working principle of the considered systems is explained by figure 1, where numbers mean: 1 - light emitter; 2 - studied objects; 3 - receiver.



Figure explaining work principle of active remote sensing.
Now we assess the value of the signal in the output of such systems. If could be shown, that signal reflected from n - th border could be assessed using formula:



where d_o - coefficient indicating fading of signal during its propagation till object; d_i - fading in i - th homogenous layer; k_i - coefficient of reflection from i - th border.
We could assume, that signals are corrected by multilication to the coefficients


In this case signal from n - th border is calculated as


Thus, corrected model fully keeps informativeness, because, information of signal reflected from n - th border is characterized by coefficient k_n.

1.3. Application of the measuring system's dimension lowering principle.
As it could be shown from formula (2) active systems of remote sensing are typical representatives of informational systems class with fading of signal. In order to synthesize an according subclass, now we consider major parameters of such systems, which could be interconnected:

^Tk_max - maximum duration of signal, reflected from mutual borders of layers;
^Tk_max - duration of holding of signal, during which signal, is faded.
Obvioully, that

^Tk_max = ^{2 T}k_max;

^Tk_min = 0;   ^Tk_min=0

As a result, averaged value of T_h could be assessed as


Thus, dimension of such a system could be lowered, if we accept that T_hav=T_k.

Now we analuse two cases of influence of energetic lossess to the informational characteristics of such systems:

1) Fading of signal is not taken into account, when number of separable grandes of signal is determined;
2) Fading of signal is taken into account during aforesaid process.
Now we prove two following theorems, according to above cases.

Theorem 1: When correction of energetic losses is lacking in a system of active remote sensing, maximum amount of information could be reached at the output of system, if two - grade regime of output signal is chosen.

Prove: We use linear equivalent system of above one. Weight function h(t), determined as reaction of a system to the input signal with one - grade amplitude is used as common charateristic.

Output signal of considered systems is determined as

U_out=U_in·h(t)

Number of grade in output signal



Where D t - time period of passing of sensing light beam through layer of an object by thickness D l. We consider the amount of information containing in received signal as a criterion of quality.



In order to investigate function (3) for extremum, we obtain its first derivative and equate it to zero. As a result we obtain following equation:

Where

Solution of transcendental equation (4) gives us x»0,8, which conforms to m=2 .
Thus, theorem 1 is proved.

Theorem 2. Maximum amount of output information could be reached in the systems of active remote sensing where correction of systemayic energetic losses (fading) is corrected, if multigrade regime of output signal is closen.
Prove: Correction of fading of received signal leads us to the fact, that number of separable grades in it could be assessed as


Where s - noise of the system, including noise of receiver.
Consequently, amount of information in the output signal could be determined as


Investigating function (5) for maximum, we obtain its first derivative on n and equate it to sero. As a result we receive following equation


Where:

For example, if y₀=30, so b and m=10.
Thus, theorem 2 is proved.

2. Optimization of Passive Remote Sensing Systems
In the second part of this submission we consider passive systems of remote sensing, where Solar radiation is used. Systematic decreasing of Solar's radition intensity in spectral band make it possible to express the spectral dependence of ratio signal/noise y as
y = y₀ + y^t(l - l₀)
where y₀=y (l =l₀);   l₀ = 3 m c m. »
In this case quantity of information is assessed as

Where l_å - parameter, characterizing total width of transparant window of passing of atmospehere, used for passive sensing within band (l-l₀);   Dl - width of the one spectral channel.

In order to syntesize the optimal system we use aforesaid principle lowering of dimension. We assume, that regularity of dependency between parameters l_å and (l-l₀), i.e. function l_å = f(l- l₀) is known. In first approach we assume linear type of said function l_å = k (l - l₀), and designating l₁ = l - l₀ we have

In order to investigate function (8) for maximum on l₁ we use above rule and obtain following equation

If we receive conditionally even distribution of noises in spectral band 3 - 5 mcm, equal to and presence of following fuction for hypothetical optical - electronic channel of radiometer in said spectral band

y(l) = 111 - 50 l₁,        (10)
then solution of (9) taking into account of (10), when y₀ = 111, y^t = 50; k=1 give us l₁ »1,7 m c m.

Thus, held analysis make it possible to conclude that there is possibility of existence of optimum spectral band width, within 3 - 5 mcm where maximum amount of authentic information is attained.

Reference

• Asadov H.H. 2000. Synthesis of one subclass of measuring systems on the basis of dimension lowering principle. Baku, Proceedings of Azerbaijan Technical University, v. VIII., ¹ 1, p. 51 - 54.
• Asadov H.H. 1982. Optimization of the process of registration on electron - beam tube. Moscow, Journal "Radiotechnics", ¹ 2, p. 64 - 66.