4 Simple Case with Volume Scattering

Transcription

1 37 4 Simple Case with Volume Scattering 4.1 Introduction Radiative transfer with volume scattering requires the solution of the radiative transfer equation (RTE), including the integral over the phase function. The general solution is very difficult, especially if the radiation is scattered many times and if the geometry is complex. So far exact solutions have been found for simple geometries and highly simplified phase functions, only. In all other situations the results are either based on approximate solutions or on numerical simulations. Nevertheless, the best insight in a physical problem is found from analytical solutions and well-defined parameters, even if the problem has to be simplified. This approach is used here. One way to simplify the RTE is by limiting the number of propagation directions, leading to a system of linear differential equations (Chandrasekhar, 196, starting at ). Most popular are Two-Stream (or Two-Flux) Models reaching back to Schuster (195), later to Kubelka and Munk (1931) and Kortüm (1969). A critical overview with a comparison of such models was given by Meador and Weaver (198), descriptions were given by Thomas and Stamnes (1999), Ishimaru (1978) and Petty (6). The simplifications lead to approximate solutions of the true RTE with an unlimited number of directions. Nevertheless, under certain circumstances the simplified model is exact or at least sufficiently accurate. Here we will present an example of an exact Two-Stream Model. The interacting waves are limited to two directions, meaning that the phase function is degenerated to a delta function. Physically the model is realised by a one-dimensional system of parallel lamellas as shown in the figure below. A main advantage is the mathematical simplicity. Although the disadvantage is the special geometry, it is interesting to note that the model produces reflectivity and transmissivity spectra that are not far different from those of real snowpacks and clouds (Mätzler, ). Indeed, thanks to the closed-form solution, the lamella-pack model can give detailed insight in the behaviour of scattering media; and the results are valuable for systems that are far more complex. Therefore we want to spend some time to investigate this model. 4. The lamella pack The model Let us assume a pack with a total height h consisting of many horizontal lamellas with a more or less fixed thickness d, whereas the separation between the lamellas is variable, determined by a mean volume fraction f of the lamella material. The scattering medium has one-dimensional geometry with the spatial variable z. z h I I1 Figure 4.1: Pack of freely arranged ice lamellas parallel to the x-y plane. Scattering couples radiation between upward (I 1 ) and downward (I ) radiation with the same incidence angle. I x

2 38 Expressing the number N of lamellas per meter depth by the volume fraction f and by the lamella thickness d, we have N = f/d (4.1) A single dielectric lamella: For a single dielectric lamella, the reflectivity and transmissivity are given by (see Equation (5.17) of the lecture notes of Mätzler (8) with Fresnel reflection coefficients F = F 1 for air to material, and material to air, respectively): 1 exp(ip) r = r 1 1 F 1 exp(ip) ; t = (1 F 1 )exp(ip) 1 F 1 exp(ip) Here r 1 = F 1 is the Fresnel reflectivity, and P is the one-way phase through the lamella ; a =1 r t (4.) P = nk dcos 1 (4.3) where nk is the complex wave number in the lamella medium, and the absorptivity a is the complement as a result of energy conservation. For lossless media, and in good approximation also for low-loss lamellas, (4.) simplifies to ( ) r = r 1 1 cosp 1+ r 1 r 1 cosp = 4r 1 sin P 1+ r 1 r 1 cosp ; a 1d <<1 cos 1 where 1 is the absorption coefficient of the lamella material, given by Equation (.7). (4.a) Smoothing interferences With increasing P, r changes between and a maximum of about 4r 1. This interference phenomenon arises from the superposition of waves reflected at the top and bottom side of the lamella. In nature d is often slightly different for different lamellas. Furthermore a radiometer usually covers a certain spectral bandwidth over which the interference is smoothed. Therefore the phase terms in (4.) are smeared out when averaged over many lamellas, except for very small values of P. Furthermore, for variable distances between different lamellas, the superposition of reflections is incoherent. If r 1 <<1, the denominator of (4.a) can be approximated by 1. The average lamella reflectivity r and absorptivity a, respectively, can be written as r = 4r 1 sin P ; P < P c ( j +1) /4, j =1,,3,... r 1 ; P P c (4.4) a = d a1 cos 1 For small average values P of P, Equation (4.4) gives the coherent reflectivity through the first maximum at one quarter of a wavelength, and it provides a continuous transition at cosp c = (for j=1,, 3,...) to the mean value, mean(sin P) =1/, from the coherent to the incoherent situation at larger thickness. We will assume j=1. Scattering and absorption coefficients For a pack consisting of N lamellas per unit depth, the scattering and absorption coefficients are the probabilities for reflection and absorption per unit depth (assuming r <<1 and a <<1) : s = Nr ; = Na (4.5) It should be noted that absorption in the host medium (e.g. air) has not been included here. See however Problem 4.

3 The transfer equation for a pack of lamellas A wave with Intensity I incident from above on the pack of lamellas (Figure 4.1) can be transmitted, absorbed or reflected, but the incidence angle (in air) is kept constant. Therefore the downwelling wave gives rise to a transmitted wave in the same direction and a reflected wave in the upwelling mirror direction (Figure 4.1). In this case we can formulate two interacting transfer equations for the up- and downwelling intensities, I1(z) and I(z). Emission is omitted here, but later it will be included through Kirchhoff's law. In analogy to Equation (.3) we write up: + di 1 dz = I 1 + s (I I 1 ) (4.6) down: di dz = I + s (I 1 I ) (4.7) Because here, the phase function is degenerated to a delta function, the term with the integral over the phase function in the radiative transfer equation is now simply expressed by the coupling terms, s I in (4.6) and s I 1 in (4.7). The first term on the right side of each equation describes absorption with an absorption coefficient, and the second term describes how much the difference K = I 1 I between the upward and downward radiation is reduced by scattering. The equations have the spatial variable z. The absorption and scattering coefficients are understood here as per unit depth z (not per path length in case of oblique incidence). Therefore the factor μ on the left-hand side of Eq. (.3) is omitted in (4.6-7). Another pair of coupled differential equations is obtained from (4.6-7) by the transformation to the sum J = I 1 + I, and to the difference: dj dz = ( + )K and dk a s dz = J (4.8) a Whereas J is the total intensity, K represents the net radiation in the upward direction whose change is not affected by scattering. The second equation corresponds to the flux equation (.6) without the emission term. The transformation to an uncoupled second-order differential equation Differentiating the first equation of (4.8) and using the second equation to eliminate dk /dz leads to the following second-order differential equation d J dz = J where = ( a + s ) (4.9) and the same equation is found, respectively, for K, I 1, I. Reflectance, transmittance and emittance of the lamella pack: All equations (4.6) to (4.9) can be solved analytically by solutions of the type I j = A j exp(+ z) + B j exp( z); j =1, (4.1) where Aj and Bj are coefficients to be determined from boundary conditions. We assume illumination from above, defining the incident radiation by I = I (top of pack). We will distinguish between two different situations, a) a semi-infinite pack at z<, and b) a pack with depth h above a non-reflecting background. Further situations could include multi-layers with different parameters in each layer or a reflecting background. a) For a semi-infinite pack (reaching to infinity on the negative z axis) the coefficient must vanish (Bj=). Then I 1 and I within the pack (z<) are proportional to exp( z). This means that the intensities

4 4 diminish exponentially with depth, and = a + s is the damping coefficient. Its inverse value is an effective penetration depth d p =1/. Inserting the Ansatz (4.1) with B1= in (4.6) gives s I = ( + s + )I 1. At the pack surface (z=) we have I = I and I 1 = r I, thus we get the reflectivity of the semi-infinite pack r = s + s + (4.11) b) For a finite depth and a non-reflecting background at z<, the boundary condition at the bottom z= and top z=h of the pack are I 1 (z = ) = ; I (z = ) = t p I I 1 (z = h) = r p I. They also define the pack reflectivity r p and transmissivity t p. The results are: r p = r 1 t 1 rt ; and t p = t 1 r 1 rt In this symmetrical pair of formulas, t is the exponential damping factor t (4.1) = exp( h) ; (4.13) and r is the reflectivity (4.11) for infinite h. Finally, the pack absorptivity a p is obtained from power conservation, and the emissivity e p follows from Kirchhoff's law e p = a p =1 r p t p (4.14) These formulas are the complete solution for any incident direction and for any polarisation. Pack without absorption Before discussing the solutions found above, it is helpful to consider the case with negligible absorption ( = ). This is called the conservative case. It follows from (4.8) that the net radiative flux K must be a constant, say K. As a consequence the equation for J must be linear with z: K = K ; J = J s K z (4.15) Transforming these quantities to I 1, I, and again using the boundary conditions I 1 (z = ) = ; I (z = ) = t p I I 1 (z = h) = r p I, we get the pack reflectivity and transmissivity in a form that looks very different from (4.1): r p = h s 1+ s h ; t = 1 p (4.16) 1+ s h Note that r p + t p =1, confirming that there is not absorption. Although most simple and reasonable, Equations (4.16) are a bit confusing. What happened to the exponential form of Lambert-Beer's law? In t of (4.13) the exponential function still appears. How can we get the transition to (4.16)? The fact that t p differs from the purely exponential form of t is indicated by Equation (4.1). The difference is negligible for small values of r, but increases with increasing r and converging to the limiting expression (4.16) for.

5 Results and discussion Physical meaning of The form of Equation (4.9) defining does not look like a common expression. Only for >> s does converge to the extinction coefficient e = + s. Note that e, the exact relation being e = + s. Our results tell us that the radiation is damped by. The reason is the enhancement of the radiation field by multiply scattered radiation. This increase can be dominant for the radiation field if x = s / >>1. Since x is the mean number of scattering events before the radiation is absorbed we can understand the propagation of scattered photons by a random walk. In this process the mean distance R by which a photon is displaced after N s scattering events is given by R = s N s where s =1/ s is the mean distance between two scattering events. Inserting N s =x and s, we get R =1/ s. This quantity can be compared with the penetration depth d p =1/ 1/ s. This is almost the same as R. The difference must be related with the one-dimensional geometry of our model. Indeed, d p refers to the z direction. In an average direction with = 45 the respective distance is increased by a factor, leading to the expression found for R. Behaviour versus depth h Note that the results (4.1-16) refer to the situation with a scattering layer of depth h above a non-reflecting background. Figure 4. shows the behaviour of r p and t p versus layer thickness h for a medium with s =1/m and for different absorption coefficients. It is apparent that has a strong influence on the results. With increasing thickness the reflectivity first increases linearly with h (the curved line arises from the semi-log representation), but saturates at large depth values. The top curve of r p still increases with h at h=1m, indicating the role of multiple scattering for small absorption. The bottom graph shows the behaviour of the transmissivity in a different semi-log representation. For large absorption the behaviour is an exponential decrease according to the Lambert-Beer law. However, the decrease of t p is not exponential if << s. This behaviour was found experimentally for light transmittance through snow (Beaglehole et al. 1998). Indeed, with decreasing the behaviour more and more approaches Equation (4.16). This result contrasts the one found for infinite depth where the exponential damping factor fully describes the z dependence of the downwelling radiation. What is the reason for this discrepancy? Note that = if there is no absorption. This means that the Lambert-Beer law can only apply to cases where absorption occurs. The actual form of the decrease of the intensity depends on the boundary condition. In our case we assumed a black body below the scattering layer. If we assumed a subsurface with a non-zero reflectivity the results would differ from (4.1-16). Especially by choosing the sub-surface reflectivity to be r the subsurface cannot be distinguished from the infinite scattering layer itself, leading to r p = r, and for the transmissivity through the scattering layer above the subsurface we get t p = t. This means that the radiation and its vertical variation inside a scattering layer not only depend on its scattering properties, but also on the surrounding medium, in other words, on the type of boundary condition. We can experience this phenomenon in nature. An example is the so-called "white-out" occurring when fog comes up while standing on a large snowfield. The radiation is completely diffuse, not allowing us to identify any preferred direction. The situation suddenly changes with the approach to an absorbing area, like deep water or dark rock.

6 4 Figure 4.: Reflectivity r (top) and transmissivity t (bottom) versus layer thickness of a volume scattering medium according to Equations (4.1) for s =1/m. The 4 curves in each graph from top to bottom are for =.1/m, =.1/m, =.1/m, and =1/m. Deep pack, and its comparison with snow For a sufficiently deep pack, t meaning that t p = and r p = r, which is only a function of the ratio x = s / = N s. From Equation (4.11) we find r = x x x 1- x x (4.17) The approximation (right-most expression) is approached for very large values of x, only. This situation is met for snow at visible wavelengths. Finding x from Equation (4.4) for large P we get x = r a r cos d (4.18) and thus from the approximation in (4.17) we find r 1 K 1 d /cos 1 (4.19) an expression for strong scattering that was used by Bohren (1987). Here, K is given by r1-1/. For vertical incidence, cos 1=1, and K is given by

7 43 n ' + 1 K = (4.) n ' 1 For n'=1.33 we get r1=. and K=7.6. According to an early snowpack model (Bohren and Barkstrom, 1974), the reflectance of a deep snowpack was written as r = D BB (4.1) where DBB is the sphere diameter for scatterers of the Bohren and Barkstrom Model. This result agrees with (4.19-) by choosing d =.61D BB (4.) Comparisons with other models, shown in the following figure, indicate slightly different factors. But the close agreement of d with the size of the scatterers in more elaborate models and also with observations gives motivation for the simple model. Figure 4.3: Decrease of the reflectance r of pure snow (wavelength = 1 μm) with increasing grain diameter D (in mm) of spherical ice grains. Data points (diamonds) along the upper curve are computed with the model of Wiscombe and Warren (198), the curve represents Equation (4.17) with x = 4.915mm/D, (d =.9D). The lower curve represents x =.69mm/D, (d =.53D) the data points were computed by Sergeant et al. (1998), using the model of De Haan et al. (1987). r grain diameter in mm At sufficiently low frequency, where the phase P is small (<1), the reflectivity increases with increasing (kd), i.e. with frequency squared. This behaviour is indeed observed for dry snow in the microwave range. This is the solution for one-dimensional "Rayleigh" scattering. The lamella pack model was used together with the dielectric spectrum of ice (Warren and Brandt, 8) to simulate spectra of a snowpack. Reflectivity (r), infinite reflectivity (r) and transmissivity (t) representative for a 1 cm pack of dry snow are shown in Figure 4.4. In addition the results of comparable reflectivities (red +) computed with a realistic microwave model are also shown. Near realistic behaviour is found over the entire frequency range from microwaves (1 GHz) to the UV (1 6 GHz). The largest reflectivity is found in the visible spectrum near the highest frequency shown in the figure because of the small absorption coefficient (i.e. large x) and large optical thickness of the snowpack.

8 44 1. t 1..8 r r r t Frequency (GHz) Figure 4.4: Radio to UV spectra of transmissivity t=t p and reflectivity r=r p of a snowpack consisting of a 1 cm deep ice-lamella pack with d=.5mm, f=.1. Also shown is the reflectivity r of the same snow, but at infinite thickness. The data points labelled + are results of a microwave emission model (MEMLS) of r for the same snow density, thickness and temperature (66K), but with a correlation length of. mm. From Mätzler. Figure 4.5: MW-to-UV Spectra at vertical incidence of transmissivity t p (red), reflectivity r p (black), and infinite reflectivity r (blue) of an ice-lamella pack representative for a thick cirrus cloud (depth 1m at an ice density of g/m 3, or 1 km with.g/m 3 ). Cloud spectra The lamella pack model was also used together with the dielectric spectrum of ice to simulate spectra of a "cirrus cloud". Results are shown in Figure 4.5. The spectral range starts at 1 GHz and reaches throughout the infrared and visible range to a wavelength of.3μ at 1 THz. It must be noted again that absorption by atmospheric gases has been ignored. Significant transmissivity (red) is found near the edge of the spectrum whereas the infrared range is opaque for ice. In contrast to Figure 4.4 the largest reflectivity (black) is found at about 1 THz. At this frequency the wavelength of.3 mm is still significantly larger than the lamella thickness. mm. The maximum is a result of a changing balance between the increase of scattering and the increase of absorption with increasing frequency. The strong

9 45 variations from 1 to THz result from the main vibration bands of the water molecule. In the visible range the reflectivity is small due to the small depth of the cloud. Note that at infinite depth the reflectivity is much higher as shown by the blue curve. Comparison with spherical scatterers Different shapes of scatterers in Figures 4.6a and 4.6b with the same specific surface show similar reflectance and transmittance spectra, see Figures a and b. Fig. 4.6a: Stack of irregularly spaced ice lamellae Fig. 4.6b: Pack of irregularly spaced spheres..9.8 Thickness=.m a: Mie, Delta-Edd. a: Lamella Model t: Mie, Delta-Edd. t: Lamella Model 1.9 Infinitely Deep Pack of Single-Size Scatterers a: Mie, Delta-Eddington a: Lamella Model Figure 4.7a: Albedo (reflectivity) a and transmittance (transmissivity) t versus wavelength ( μm) of a cm deep snowpack, density 1kg/m 3, incidence angle 53. Comparison of spheres (D=.8mm) with lamella pack for lamella thickness d=d/3; blue: Mie-Delta- Eddington model of Wiscombe and Warren (198), black: lamella model of Mätzler (). Figure 4.7b: As Figure a, but for a deep snowpack with illumination adapted to the respective geometry: vertical incidence for lamella pack, diffuse illumination for pack of spheres. The spectral albedos of the two situations agree almost perfectly. Problems 1) Plot the reflectivity r of a halfspace versus x = s / from very small (<x<<1) to very large (x>>1) values, and interpret the results. ) Compute values of r p and t p according to (4.16) and plot them in Figure 4.. Can you see a difference with the curves of the lowest value, and if yes, where?

10 46 3) Find expressions for r p and t p of the subsurface below the lamella pack has a non-zero reflectivity r s. For simplicity assume specular reflection to avoid the mixing of radiation at different incidence angles. See Chapter 13.7 of Petty (6) for a similar situation. 4) How would the results of Equations (4.1) change if the host medium were slightly absorptive, e.g. an ice cloud where absorption in air cannot be neglected? Describe in words how you would adapt the present model. 5) Read Chapter 13 of Petty and find the correspondence to the present model. 4.5 MATLAB functions function result = lamella(fghz, thetad, epsilon, d, nphase) % Transmission, reflection and absorption of EM waves on a plane, % dielectric lamella in vacuum. Coherent computation for % phase<ppp else incoherent. % Literature: Kong, J. A. (1986). Electromagnetic Wave Theory. % Adams, R. N. und E. D. Denman (1966): Wave Propagation % and Turbulent Media, American Elsevier % Input: % fghz: Frequency [GHz], thetad: Incidence angle [deg], % epsilon: complex, relative dielectric constant % d: thickness of the lamella [m] % nphase=1,,.. number of halve waves+1/4 with coherent computation % uses fresnel( ) function result = lamella3(lamda, d, thick, mu, mv) % Radiative transfer for unpolarised ice-lamella pack % Input: % lamda: Wavelength in micron, d lamella thickness in m % thick= slab thickness in m, mu=cos(incidence angle) % mv ice-mass density in kg/m^3 % uses lamella(.,.,.,.,nphase=1) % Output: r, r, t (mean values of h and v pol.) function result = lamellaspec(fmin, fmax, nla, mu, d, thick, mv) % Logarithmic spectral plot (THz) of unpolarised lamella pack % Input: % fmin, fmax: minimum and maximum frequency in THz, % nla: number of frequencies, mu=cos(teta), teta=incidence angle % d: lamella thickness in m, mv: pack density in kg/m^3 % thick: slab thickness in m function result = lamellaspec(lamin, lamax, nla, mu, d, thick, mv) % Spectral plot of unpolarised lamella pack % Input: % lamin, lamax: minimum and maximum wavelength in micron, % nla: number of wavelengths, mu=cos(teta), teta=incidence angle % d lamella thickness in m, mv: pack density in kg/m^3, % thick: slab thickness in m function result = lamella1(ks,ka) % Plot of reflectivity versus thickness of a lamella pack using % the two-flux model % Input: % ks,ka: scattering, absorption coefficient, 1/m function result = lamella(ks,ka) % Plot of transmissivity, otherwise the same as lamella1