Radio Propagation and Remote Sensing of the Environment - Chapter 8 pdf

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Radio Propagation and Remote Sensing of the Environment - Chapter 8 pdf

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© 2005 by CRC Press 221 8 Radio Thermal Radiation 8.1 EXTENDED KIRCHHOFF’S LAW The background of the thermal radiation theory of heated bodies will be discussed in this chapter. This radiation appears as the result of random motion of charged particles inside the body. The velocities of this movement are stochastic and, in particular, they change their value and direction occasionally as a result of the interaction (collisions) of particles with each other. The radiated field strength is random, and its intensity depends on the particle energy and, consequently, on the temperature of the body. In this connection, the radiation under discussion is referred to as thermal . We have to imagine that these bodies, and the fluctuation field generated by them, are in a giant thermostat that maintains the thermodynamic balance. This means that the charged particles of the body interact with the given fluctuation field, derive energy from it, reradiate it afresh, and then absorb, reradiate, and so on. In a word, the radiating and absorbing energies are balanced in an equilibrium state for the fluctuation field. The fluctuation field itself can be described as the field radiated by random currents with density j ( r ). The mean value of this density is equal to zero, and the spatial correlation function of its frequency spectrum is defined on the basis of the fluctuation–dissipation theorem (FDT): 24,56 (8.1) Here is Planck’s constant. The subscripts α and β represent corresponding coordinate components of the current vector. The FDT, described in such a way, is correct over practically the entire electromagnetic spec- trum (at least, for wavelengths that exceed interatomic or intermolecular distances). The energy quantum in the radio region is small (i.e., the inequality is true). Therefore, we will use this approach when the averaged energy of quantum oscillator: (8.2) j,j coth b αβ ωω ω π ω ε ′ () ′′ () =       ′′ rr,  2 2 8 2kT ωωδ δ αβ () ′ − ′′ () rr .  ==⋅⋅ − h 210510 27 π .secerg ω << kT b Θ(,)ω ωω T kT =        22 coth b TF1710_book.fm Page 221 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 222 Radio Propagation and Remote Sensing of the Environment is substituted for its approximate value and (8.3) is valid in the radiofrequency band. It is characteristic that orthogonal components of the fluctuation current are not correlated. For the current components themselves, the spatial correlation radius in this case is equal to zero. In fact, it has the scale of particles interaction — for example, interatomic distances in a solid body or free path length in a gas. Because the wavelengths considered here exceed these distances, it is possible to neglect their variation from zero. The most important fact is that the spectral density of the fluctuation current is defined by the imaginary part of the body permittivity (i.e., its ability to absorb electromagnetic waves). Kirchhoff’s law applies here implicitly, as it connects radiating and absorbing properties of the body. Let us point out in this connection, that we are dealing with nonmagnetic materials, so the magnetic losses default, and we do not need to represent the magnetic fluctuation currents. In order to calculate the fields generated by fluctuation currents, we need to know Green’s function of the considered body — that is, the diffraction field excited inside the body by a single current source: (8.4) where e is a single vector, generally speaking, of arbitrary direction. The field , excited by this current, is the diffraction field. To determine the fluctuation field, we can use the mutuality theorem in the form of Equation (1.64). The fluctu- ation current and field is represented by , while represents the current (Equation (8.4)) and the diffraction field generated by it. Omitting unnecessary subscripts, we now have the following equation for the fluctuation field: (8.5) Also, we have the expression for the calculation of the fluctuation field component, oriented in the direction of vector e . Its average value is equal to zero, as the average value of the fluctuation current is also equal to zero. In this connection, let us point out that the diffraction field is the determining value. Let us also emphasize another very important fact. The imaginary part of the dielectric permittivity in Equations (8.1) and (8.3) can be a function of the coordinates. Particularly, it can be equal to zero if, for example, part of the considered volume V occupies a vacuum. So, the volume can include as the actual heated body, which serves as a fluctuation field source, any part of space up to infinity. It is important that point r of the dipole, existing in the diffraction field, is situated inside the volume, but it can be outside Θ = kT b , jj bαβ αβ ωω ω π εω δ δ,, ′ () ′′ () = ′′ () ′ − ′′ () rr rr 4 2 kT jr err G ′ () = − ′ () δ , EH dd , jE 11 , jE 22 , eEr jr E rre r⋅ () = ′ () ⋅ ′ () ′ ∫ d V d,, . 3 TF1710_book.fm Page 222 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Radio Thermal Radiation 223 of the material body. In particular, it can be removed from the radiating body so far that the incident on the body wave is practically a plane wave. Due to the isotropy of the fluctuation currents and statistical properties and based on the FDT, we now have the following for the mean intensity of the field component: (8.6) If we now recall Equation (1.20), describing the density of losses of electromagnetic energy in a substance, then the last result can be rewritten as: (8.7) So, the intensity of the fluctuation field is determined by the value of the thermal losses of the diffraction field excited in the body by a unit current applied at the point where the fluctuation field is being studied and directed according to the vector of the fluctuation field polarization. The result obtained is sometimes referred to as the extended Kirchhoff’s law. It is called extended because the law initially formulated by Kirchhoff was restricted to the case of a body large in size compared to the wavelength (i.e., the geometrical optics problem). No such limit is stated in the relations being considered here, so the extended Kirchhoff’s law applies. We should point out the dependence of the integrands in the previous formulae on vector e and the diffraction field dependence on the auxiliary dipole polarization; thus, the polarization is dependent on the radiation of the heated bodies. Let us suppose now that the radiation is detected by a receiver responding to only one linear polarization. We can assume that the receiver is rather distant and detects the waves with polarization orthogonal to the line connecting the receiver and the center of gravity of the radiating body. We can direct the z-axis along this line. The discussion in this case is about the reception of waves, the polarization of which is oriented in the plane {x, y}. Let us consider the case of a receiver detecting the x- polarization. In this case, it will react not only to the x-polarization waves but also to waves polarized in the plane. The difference between these waves and the x- polarized waves is that the power of their fluctuations will be detected by the receiver with the weight , where η is the angle between vectors e and e x . In the case of statistical independence of waves of different polarization, the fluctuation intensity of the detected x-polarized field will be equal to the weighted sum of intensities of all the fields polarized in the {x, y} plane. In other words, (8.8) eEr rErre r⋅ = ′′ ′ () ′ () ′ ∫ () , , , . 2 2 2 3 4 ω π εω kT d V b d eEr rre r⋅ = ′ () ′ ∫ () , , . 2 3 2kT Qd V b π cos 2 η E x b 2 23 0 2 2 = ′ () ′ ∫∫ kT Qdd V π ηηη π rr r,, cos . TF1710_book.fm Page 223 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 224 Radio Propagation and Remote Sensing of the Environment In the case of the receiver responding only to the y-component of waves, we have: (8.9) If the receiver detects both orthogonal polarizations, then the power of the resulting field will be equal to the sum of Equations (8.8) and (8.9). Let us now compute the spectral density of Poynting’s vector z-component for different polarizations. To do so, we must take into account that the spectral densities given by Equation (8.7) and others are two-way (i.e., they are applicable over the entire real axis of frequencies from – to +. We, however, are interested in a one-way spectral density that is associated with the positive half-axis of frequencies. This means that the previous expressions should be multiplied by 2: (8.10) 8.2 RADIO RADIATION OF SEMISPACE Let us now consider the radiation of semispace. We will first consider the case when the receiving antenna is situated in a vacuum and directed perpendicularly to the plane boundary of the radiating medium. In this case, the diffraction field and, consequently, the volume density of absorption do not depend on the polarization. The integration in Equation (8.10) can be performed over the angle to give the factor π . Further, we are interested in the spectral density of the power flow that is detected by the receiver for any linear polarization: . (8.11) Here, is the antenna area, and integration with respect to s represents integration over the plane that is perpendicular to the z-axis. Writing the expression in such a form shows that we have implicitly used the geometrical optics approach. Equation (8.11) assumes that the radiation field in the view of plane waves coming from different directions reaches the antenna and summarizes their intensities with a weight given by the antenna area value. One can point out, in this connection, that this approximation requires the position of the point in the field being searched to be at a distance from the interface much greater than the wavelength. Equation (8.11) E y b 2 23 0 2 2 = ′ () ′ ∫∫ kT Qdd V π ηηη π rr r,, sin .   S S 2 E E x y x y ω ω π () ()           =            c 2 2    = ′ ()           ck T Qdd b π η η η η 2 2 2 rr,, cos sin 33 0 2 ′ ∫∫ r . V π  W ck T Ad d ex b d Ez) z() () () (,ω ωεω π = ′′ ∞ ∫ 8 2 2 2 2 0 ss s ∫∫ A e TF1710_book.fm Page 224 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Radio Thermal Radiation 225 implies at the same time that the antenna reacts only to polarization orthogonal to the z-axis, which occurs only in the case of a highly directional antenna. Therefore, the main area of integration with respect to s in Equation (8.11) is concentrated close to the coordinate origin. From here, using the geometrical optics approxima- tion, the diffraction field can be expressed as: where is the field of auxiliary dipole on the interface and F (0) is the Fresnel reflective coefficient at zero incident angle. After integration over z and simple transforms, we obtain: The field on the surface is easily calculated if we take into account that it is generated by an electrical dipole with moment equal to the value (compare with the first term of Equation (1.40)). The dipole field on the surface being considered close to point s = 0 is: (8.12) according to Equation (1.38). Further, we should keep in mind that (the solid angle element) and use Equation (1.122). As a result, . (8.13) Just the same result will apply to a y-field component, so the spectral density of the total power flow can be written as: (8.14) This result is obvious and reflects the detailed balance between emitted and reflected energy flows. It is convenient to express the power detected by an ideal receiver (i.e., by a receiver with perfectly matched circuits) in the temperature scale. Such temperature is called brightness and is equal to: EzEzE dd z s,, , () ≅ () ≅ + ()     010 0 i ik Fe ε E i 0  W ckT F Ad ix 2 b e E= − () ∫ 10 8 2 2 0 2 2 () () . π ss p = −1 iω Ee i ik k i e 0 2 = ω r p r dd 2 s r = Ω  W kT F x b () ()ω = −       2 10 2  WFkT() () .ω = −       10 2 b TF1710_book.fm Page 225 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 226 Radio Propagation and Remote Sensing of the Environment (8.15) Note that our method of calculation leads to the-so called antenna temperature; however, there is no difference between brightness and antenna temperature in the considered case of semispace. The black-body reflective coefficient is equal to zero and, in this case, the brightness temperature is simply equal to the temperature of the black body. So, the brightness temperature is the temperature of a black body at which it radiates with the same intensity as the heated body at a given polarization and frequency. In the example discussed here, the following value is the emissivity (coefficient of emis- sion): (8.16) and it is more convenient to write: (8.17) This last expression is considerably more widely used than the particular case from which it was obtained. At the inclined observation of a plane-stratified medium, the emissivity is equal to: . (8.18) Here, the second summand is equal to half the sum of the reflective coefficients for the horizontal and vertical polarized waves. These coefficients for a plane-layered medium differ, in general, from the Fresnel ones. The angle θ is the zenith angle of observation in this case. Let us emphasize the fact that Equation (8.18) concerns media inside which radiation that has penetrated is fully absorbed. So, for example, Equation (8.18) has to be modified by adding the transmission coefficient in the case of a limited-thickness layer. In the general case, (8.19) Let us point out two circumstances regarding the radiation of a semispace filled by a transparent, or more exactly, a weakly absorbing medium. First, the brightness temperature is equal to the temperature of the body observed on vertical polarization at the Brewster angle. This means that the medium is like a black body (the emissivity is equal to unity) under the specified conditions. If we were to perform the vertical TFT  () .ω = − ()       10 2 κω() ()= −10 2 F TT  () () .ωκω= κωθ θ θ(,) () ()= − +       1 1 2 22 FF hv κωθ θ θθθ, iiiii FFTT () = − () + () + () + ()     1 1 2 h vhv TF1710_book.fm Page 226 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Radio Thermal Radiation 227 polarization emission measurement and change, in the process, the angle of obser- vation, then the body temperature is determined at the angle of maximum radiation, and the dielectric constant is calculated by this angle value. The second circumstance is connected with Equation (3.25), from which it follows that the temperature of an emitting semispace can be easily determined by observing both polarizations at θ = 45°. It is computed by the formula: (8.20) The emissivity is a function of frequency. This frequency dependence is twofold due to the permittivity frequency dependence and to the interference and resonance phenomena that are described by the diffraction field. The temperature of natural media cannot be constant over the space. This situation is typical, for example, for soil, which is not uniformly heated by solar radiation. In these cases, the formulae defining the brightness temperature demand elaboration. One should take into account when performing the corresponding cal- culations that, strictly speaking, a medium that is not uniformly heated is not in equilibrium, even with the heat transfer process; however, if spatial temperature gradients are rather small, then the medium can be assumed to be locally in equi- librium. The definition of small gradients is not formulated in the general case and always requires elaboration, taking into account the peculiarities of the problem being studied. One can assume, in our cases, that the demand of small spatial gradients of temperature is always fulfilled. The temperature spatial variations are followed by a spatial change of the heated medium permittivity, as the permittivity is a temperature function. The spatial changes, in this case, should also be taken into account. Further, we will summarize the problem and take into consideration permittivity spatial changes caused by various factors, not only temperature. Among these are spatial variation of the medium density or concentration changes in impurities. Let us examine, for instance, the case of a semispace that is not uniformly heated (e.g., soil in the morning). The permittivity of the medium will be assumed to be a function only of depth, and we will concentrate on observation at the nadir. Instead of Equation (8.11), we now have: (8.21) Let us assume that the temperature and permittivity change slightly on a scale of the order of the wavelength. We can use, in this case, the Wentzel–Kramers–Brillouin (WKB) approximation for the diffraction field. According to this, the reflection inside the medium cannot be taken into consideration, and we should analyze only the T T TT = () −   (h) (h) (v) 2 2 . T c Ad T d e = ′′ ∞ ∫∫ ω π ε 4 z) z) E z) z. 2 d () ( ( (,ss s 2 2 0 TF1710_book.fm Page 227 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 228 Radio Propagation and Remote Sensing of the Environment wave reflection on the medium–vacuum interface, which is characterized by the reflective coefficient: As a result, (8.22) where the coefficient of absorption is: (8.23) After rather simple transforms, we obtain: (8.24) Let us now regard two extreme cases. The first one refers to weak absorption in the sense that the imaginary permittivity part is much smaller than its real one; that is . Then, we will neglect the imaginary permittivity part everywhere it is reasonable to do so to obtain: (8.25) In the case of strong absorption, when the real and imaginary parts of the permittivity are comparable, the integral from the absorbing coefficient changes quickly on the wavelength scale. Other cofactors in Equation (8.24) can be assumed to be “slow” functions, which gives us the opportunity, using integration by parts, to obtain an expansion with respect to the reverse degree of ΛΓ (0), where Λ is the scale of the temperature or permittivity change. Because ΛΓ (0) ≈ l/ λ in this case, the series terms decrease quickly. The sum of the first two is represented in the form: F = − + = 1 1 0 0 0 0 ε ε εε,(). E r z d 2 z 2 4 2 2 0 0 1 = + () − ()         ∫ kF d ω ε ε ζζexp ,Γ Γ((((.z) z) z) z)== −       = ′′ ∗ 22γεε k i kn TF T d  = − () + () () () () − () ∗ 1 2 1 2 0 εε ε ζζ zz z zz z ΓΓexp ∫∫∫         ∞ . 0 ′′ << ′ εε TFT dd  = − () − ()         ∫ ∞ 1 2 00 ((expz) z) z. z ΓΓζζ ∫∫ TF1710_book.fm Page 228 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Radio Thermal Radiation 229 (8.26) It follows from Equation (8.26), that, at strong absorption, the temperature spatial variability is not appreciably displayed in the intensity of thermal radiation. It is understandable, because in this case the essential contribution to the radiation is brought by the fluctuation currents that are situated close to the medium interface. Let us now consider the case of an eroded boundary of radiating medium to see what corrections the unsharpness of border introduces into the emissivity. We will assume that thickness d of the transient layer is small in comparison with the wavelength. In order to calculate the emissivity at the zero incident angle, we will use Equation (3.119) to obtain: (8.27) We also must use the model for the permittivity depth distribution: (8.28) The advantage of this model is that the function describing the permittivity depth profile is integrated with the points z = 0, d and has zero derivatives there. For this reason, it “is smoothly connected” with the permittivity values on the boundaries of the transient layer. Integration over the equations provided in Section 3.8 gives us: (8.29) It is easy to see that, as was expected, the correction is small at the accepted approaches; however, we can also see that it increases the emissivity. It takes place because of better matching between the vacuum and the emitting medium. The transient layers improve the matching rather appreciably in some cases, as can be seen by analyzing Equation (3.41). Without going into the details, let us consider the case of dielectric media under the conditions of , where Let us now set the conditions at which the reflective coefficient of the layer system converts to zero. First, the condition should be observed in order to change the cosine in the numerator of Equation (3.41) to –1; in fact, this TFT d d T  = − () () + () () + ()          ∗ 101 1 0 2 Γ z z εε ε                =z0 . κω() () .= − () − + − ()       {} 1121 0 22 2 1 2 FkFIFJ fff εε ε(.z) zz dd =+ − ()       −− ()       13 1 2 1 23 dd κω ε () = − () + − ()           11 1 20 2 2 22 F kd d f . 1 23 <<εε ψψπ 12 23 ==. 2ψπ= TF1710_book.fm Page 229 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 230 Radio Propagation and Remote Sensing of the Environment will take place at any odd numbers of π . Here, however, we will restrict ourselves to a very simple case. Under these conditions, we are dealing with a quarter-wave layer of thickness (a quarter of the wavelength in the layer). Because we are discussing layers for which the thickness is on the order of the wavelength as well as weakly absorbing media, then we should assume that τ = 0 in Equation (3.41). Then, the second requirement of converting the reflective coefficient to zero and, correspondingly, the emissivity to unity will be satisfied by the equality which leads to the necessary validity of the relation We should emphasize the resonant character of the effect described, as the monochromatic emission is under discussion. Because reception of thermal radiation takes place in the frequency bandwidth, which is often a rather wide one, then the mentioned resonance can be eroded and full conversion of the emissivity to unity does not occur. This erosion is found to be weak in the case of a narrow bandwidth. The corresponding analysis should be carried out using, for example, Equation (3.57). In the case considered here, . Then, by expansion in a series, the following is obtained: (8.30) Here, f 0 is the central frequency and ∆ f is the bandwidth of the receiving frequencies. It would appear that the difference between the quarter-wave layer emissivity and unity seems to be small. Let us consider, finally, the radiation of a weakly reflected layer. The diffraction field can be described by the WKB approximation in this case; therefore, the expres- sion for the brightness temperature is obtained from Equation (8.25), where the reflective coefficient is set equal to zero (i.e., no sharp jump of permittivity on the border). So, (8.31) Here, d is the layer thickness. In the case of constant temperature inside the layer, (8.32) d = λε4 FF 23 12 = , εε 32 2 = . ββ 21 1− << κ πε ε = −≅− − ()       11 1 192 2 2 2 2 20 2 F f f ∆ . TT dd zd  = () () − ()         ∫∫ zz zΓΓexp .ζζ 00 TT e d d  = − () = () − ∫ 1 0 τ τζζ,.Γ TF1710_book.fm Page 230 Thursday, September 30, 2004 1:43 PM [...]... sizes much smaller than the antenna pattern width In this case, the bodies have to be in the wave (Fraunhofer) zone of the antenna, and the antenna itself is in the Fraunhofer zone of the emitting body; therefore, the field of auxiliary dipole in the area of this body coincides to a high degree of accuracy with the plane wave field Then, if one compares Equations (5.9), (8. 6), and (8. 7), it is a simple... 2004 1:43 PM 232 Radio Propagation and Remote Sensing of the Environment We will now introduce the black body concept for this case Let us project the radiating body onto the plane (Figure 8. 1) that is perpendicular to the line linking the radiating body and the antenna (in our case, the z-axis) We will use Σ(e i ) to represent the square of the projected body Obviously, the value of this projected... PM Radio Thermal Radiation 231 If the integral absorbing coefficient (optical thickness) τ is large, then the brightness temperature is equal to the temperature of the layer, and the latter temperature becomes similar to that of the black body In the case of small absorption, T = τ T; that is, the brightness temperature is proportional to the integral absorption in the layer 8. 3 THERMAL RADIATION OF. .. estimation of the emissivity on the basis of Equation (8. 7) The corresponding calculation for the small roughness case is very simple We will restrict ourselves to studying the vertical incidence of the single amplitude plane 2 wave The reflected energy is determined by the power flow c F(0) 8 From this, we should subtract the flow of the second approximation coherent component and add the flow of the first... roughness and large roughness To estimate the emissivity value, we will use a technique based on the problems of wave scattering that were solved in Chapter 6 which allows us to determine the amount of energy absorbed by the semispace In order to do this, it is sufficient to calculate the amount of energy reflected and scattered by the interface The remaining energy enters the body and is absorbed there The. .. of radiation of bodies with roughness requires special consideration It is natural, as the scattering processes add up to the reflection and they make their own contribution to the energy exchange between a body and a field © 2005 by CRC Press TF1710_book.fm Page 234 Thursday, September 30, 2004 1:43 PM 234 Radio Propagation and Remote Sensing of the Environment We do not intend to study the general problem...  ε ε  ΣA 1+ µ (8. 61) ∫ Further, we can assume small slopes and perform the necessary expansions and approximations to obtain: ( ) κ v,h = 1 − Fv,h θ i © 2005 by CRC Press 2 + ∆κ v,h , (8. 62) TF1710_book.fm Page 240 Thursday, September 30, 2004 1:43 PM 240 Radio Propagation and Remote Sensing of the Environment The corresponding corrections for roughness differ for the two types of polarization For... FIGURE 8. 1 Projecting a body on the plane ( ) Σ (e ) σ (ax,y) ω , η (8. 36) i ( ) ( ) Further, note that Σ e i r 2 = Ω e i is the solid angle at which the considered body can be seen from the center of the antenna at this direction, and we have: Ae ( 0 ) = λ2 , ΩA (8. 37) where Ω A is the angle spread of the antenna pattern As a result, we now have: ( ) T ( x,y) ω , e i = T ( )κ Ω ei ( x,y) ΩA (8. 38) Let... Thermal Radiation 233 The integrals on the right-hand side perform the summation of emitted particles over the volume with regard to the fact that their radiation falls within different areas of the antenna pattern We have taken into account the particle radiation attenuation due to extinction upon its propagation in the scattering medium This is the reason why our statement about the particle emission... separation has a local character and changes from point to point The incident field is set by Equation (8. 12) We can now introduce a local coordinate system consisting of the vector of the wave incident (ei); the orthogonal to ei and tangent to the surface vector (n × e ) i ( ) × (n × e )   ( ( ( ) ) 2 1 − n × e i ; and the orthogonal to ei and lying in the local plane of the ) 1− n × ei incident vector . by CRC Press 224 Radio Propagation and Remote Sensing of the Environment In the case of the receiver responding only to the y-component of waves, we have: (8. 9) If the receiver detects. 2005 by CRC Press 222 Radio Propagation and Remote Sensing of the Environment is substituted for its approximate value and (8. 3) is valid in the radiofrequency band. It is characteristic. 236 Radio Propagation and Remote Sensing of the Environment Let us represent the incident wave polarization vector in the form: (8. 46) The fluctuation field received is divided into the field of

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  • Table of Contents

  • Chapter 8: Radio Thermal Radiation

    • 8.1 EXTENDED KIRCHHOFF’S LAW

    • 8.2 RADIO RADIATION OF SEMISPACE

    • 8.3 THERMAL RADIATION OF BODIES LIMITED IN SIZE

    • 8.4 THERMAL RADIATION OF BODIES WITH ROUGH BOUNDARIES

    • References

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