electromagnetic simulation of radar signals

MASTER 2 RECHERCHE OR INTERNSHIP REPORT

Nikola BEŠIĆ

ELECTROMAGNETIC SIMULATION OF RADAR SIGNALS: APPLICATION ON STUDIES OF SNOW

Supervisor: Gabriel VASILE

Chargé de Recherche CNRS - Grenoble INP, Gipsa-lab

Tutor: Yannis LE GUENNEC

Maîtres de conférences – Grenoble INP, IMEP-LAHC

President of the jury: Jean Daniel ARNOULD

Maîtres de conférences – Grenoble INP, IMEP-LAHC

Grenoble, June 2011.

M2R OR Internship Report

Nikola BEŠIĆ

Acknowledgments First of all, I would like to express gratitude to my supervisor Dr. Gabriel Vasile, who has invested a lot of efforts to introduce me this subject and has helped me significantly with the advices while I was conducting my research, during the previous fourth month period. Discussions with him were essential for my problems comprehension. I also must mention that in the very last phase, without his suggestions concerning the improvement of my writing style, this report wouldn’t be very clear and understandable. I sincerely hope we will continue working together on my thesis. I must also thank to my tutor Dr. Yannis Le Guennec, whose comments and advices after revision of my bibliography report were very useful for the continuation of my stage. Considering my problems with the French language and by that with staying well informed about all the procedures, I really appreciate his willing to provide me all the necessary information concerning my responsibilities toward the school all the time. I own gratitude to the staff of the GIPSA lab, especially to my friends and colleges with whom I was sharing the office: Fatima, Paula, Laura, Mathilde, Radu, Fakhri, Victor and George for making my time spent there extremely pleasant and unforgettable. The last, but not the least important, I would like to thank to my father Mr. Mladen Bešić, who has helped me a lot with a graphical design, while I was preparing this report. Also, even though engineer in totally different field, he was able to give me some quite valuable advices concerning mathematical problems I was facing.

Nikola Bešić, Grenoble, June 2011.

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Contents

Introduction .................................................................................................................................... 7 I General context ........................................................................................................................... 9 I.1 Snow cover.................................................................................................................................. 9 I.2 Active microwave remote sensing ............................................................................................ 10 I.2.1. Synthetic Aperture Radar (SAR)....................................................................................... 11 I.3 Radiative transfer theory ........................................................................................................... 13 I.4 Image analysis ........................................................................................................................... 14

II Electromagnetic properties of a snow cover ................................................................. 17 II.1 Dielectric properties................................................................................................................. 17 II.1.1 Dielectric constant............................................................................................................. 17 II.1.2 Dielectric constant of a snow ............................................................................................ 17 II.2 Backscattering mechanism ...................................................................................................... 19

III Dry snow analysis ................................................................................................................. 23 III.1 Theoretical introduction ......................................................................................................... 23 III.1.1 Rayleigh scattering model ............................................................................................... 23 III.1.2 QCA model ...................................................................................................................... 25 III.2 Sensitivity analysis ................................................................................................................. 28 III.2.1 L-band .............................................................................................................................. 28 III.2.2 C-band ............................................................................................................................. 33 III.2.3 X – band ........................................................................................................................... 36 III.3 Statistical approach................................................................................................................. 39 III.4 Semi – empirical model.......................................................................................................... 41 III.4.1 Dry snow density estimation using L-band data.............................................................. 42 III.4.2 Dry snow depth estimation using C and X band data...................................................... 43

IV Wet snow analysis ................................................................................................................. 49 IV.1 QCA – CP model ................................................................................................................... 49 IV.2 Simulation analysis ................................................................................................................ 51 IV.2.1 L-band ............................................................................................................................. 51 IV.2.2 C band.............................................................................................................................. 54

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IV.2.3 X-band ............................................................................................................................. 56 IV.3 Multilayer wet snow cover ..................................................................................................... 58

Conclusions and Perspectives ................................................................................................. 61 BIBLIOGRAPHY ....................................................................................................................... 63 Appendix ........................................................................................................................................ 65 Simulator codes............................................................................................................................ 75

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Introduction Snow cover can be defined as a layer composed of translucent ice crystals. Among with sea ice, freshwater ice, land ice and frozen ground, it constitutes the Earth’s cryosphere. Snow presence is inversely proportional to temperature which implies that most of it is spatially distributed near Earth's poles and at the high altitudes. This makes its observation suitable for remote sensing, because of the numerous difficulties in doing “in situ” measurements. Snow cover analysis is important due to the several reasons. At a global scale, snow cover has a significant influence on global climate system and hidrology. Former, through controlling Earth's albedo value, thus heat quantity absorbed by the ground, and serving as a thermal insulator which reduces heat exchange between ground and the atmosphere while later, through the melting effect, especially in case of a seasonal snow cover. At a local scale, it serves as a source of drinking water, and which is particularly important for our project, as a source of cooling water used in nuclear power plants. Remote sensing of the Earth is defined as the science of acquiring information about the Earth's surface without being in contact with it, by sensing and recording reflected or emitted electromagnetic energy and processing, analyzing and applying that information. In order to extract useful information from collected data, it is necessary to know the way electromagnetic wave interact with appropriate target terrain, in our case, with a snow cover. In other words, it is necessary to know how parameters of the observables affect scattered EM wave. From practical point of view, one of the most useful snow cover parameter is snow water equivalent (SWE), so our goal during the previous period was to analyze, using fundamental theories, influence of physical parameters, which are defining SWE, on snow cover backscattering coefficient. This way, we managed to theoretically justify, by doing comprehensive sensitivity studies, already existing semi-empirical models which can be used in real data analysis, and to provide some directions and basis for further modifications of those models and creation of the new ones. Results of our theoretical analysis can also be used as the guidelines for future acquisition parameters. This report is organized as follows. First chapter is providing basic introduction to active microwave remote sensing technique, radiative transfer theory and snow cover properties, necessary in order to completely understand theories and simulation procedures and results, which are given in the third and fourth chapter. There is also a part dedicated to the image analysis, where images which are going to be used in validation process are introduced. Aside simulations descriptions and results, third and fourth chapter are also providing conclusion concerning, respectively, dry and wet snow backscattering, followed by physical interpretations. In the second chapter, we are treating snow cover, from the electromagnetic point of view, through its dielectric properties and through the description of backscattering mechanism.

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I General context The first chapter provides brief description of the target, which is in this case snow cover, with elaboration of all the important parameters which physically define it, along with a SAR remote sensing principal concept. It also introduces radiative transfer theory, which represents the core of the remote sensing target electromagnetic modelling. At the end, short part concerning SAR image analysis with the example of the old acquisition result is exposed.

I.1 Snow cover In general, snow cover represents a mixture of ice crystals, liquid water and air. The proportion of these materials is a function of many factors, but the most dominant one is temperature. For temperatures below 0 ○C, the proportion of liquid water is negligible so we call that kind of snow – dry snow. Above this temperature, depending on some other factors, certain quantity of liquid water can be present, in which case snow is characterized as a wet snow. The most basic physical parameter of a snow pack is the density, expressed in kg/m3: 𝜌𝜌𝑠𝑠 =

𝑚𝑚𝑖𝑖 + 𝑚𝑚𝑤𝑤 𝜌𝜌𝑖𝑖 𝑉𝑉𝑖𝑖 + 𝜌𝜌𝑤𝑤 𝑉𝑉𝑤𝑤 = = 𝜌𝜌𝑖𝑖 𝑓𝑓𝑖𝑖 + 𝜌𝜌𝑤𝑤 𝑓𝑓𝑤𝑤 𝑉𝑉𝑠𝑠 𝑉𝑉𝑠𝑠

(1.1)

where 𝑚𝑚𝑖𝑖 and 𝑚𝑚𝑤𝑤 are representing masses of ice and water, 𝑉𝑉𝑖𝑖 , 𝑉𝑉𝑤𝑤 and 𝑉𝑉𝑠𝑠 are volumes of ice, water and snow, respectively, 𝜌𝜌𝑖𝑖 (917 kg/m3) and 𝜌𝜌𝑤𝑤 (1000 kg/m3) are appropriate densities, while 𝑓𝑓𝑖𝑖 and 𝑓𝑓𝑤𝑤 are volume fractions. Typical density values for Alpine region, according to Météo France measurements are from 200 kg/m3 for dry snow to 600 kg/m3 for wet snow layer. Principally, due to the compaction caused by wind, gravity and thermal metamorphism, density is increasing with time according to: 𝜌𝜌𝑠𝑠 = 𝜌𝜌0𝑠𝑠 (1 + 𝑡𝑡)0.3

(1.2)

The very purpose of our project is to estimate, the snow cover derived parameter – snow water equivalent (SWE). It is defined as the depth of the layer of liquid water that would be produced if all the ice in the snow pack were melted (1.3). 𝑑𝑑

1 𝑆𝑆𝑆𝑆𝑆𝑆 = � 𝜌𝜌𝑠𝑠 𝑑𝑑𝑑𝑑 𝜌𝜌𝑤𝑤

(1.3)

0

In order to determine SWE it is necessary to know snow layer depth and its density. Those two variables are not highly correlated, so usually, as it will be shown, different methods are used for their estimation. Except density and depth there is one more, in the context of remote sensing, very important physical characteristic of a snow cover – snow surface. The reason is that the total backscattering from snow can be significantly affected by the snow surface backscattering component, especially in the wet snow case.

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Snow surface, like any other surface is described using root mean square (RMS) height deviation of a surface and surface correlation length. The root mean square height deviation 𝜎𝜎 is given by:

𝜎𝜎 2 = 〈(ℎ(𝑥𝑥, 𝑦𝑦) − 〈ℎ(𝑥𝑥, 𝑦𝑦)〉)2 〉 ≡ 〈(ℎ(𝑥𝑥, 𝑦𝑦))2 〉 − 〈ℎ(𝑥𝑥, 𝑦𝑦)〉2

(1.4)

where ℎ(𝑥𝑥, 𝑦𝑦) represents a surface height at given coordinates.

Surface correlation length 𝑙𝑙𝑐𝑐 is defined using autocorrelation function for one dimension (𝑥𝑥), which shows how the surface profile is similar to itself when it is displaced for 𝜉𝜉 in 𝑥𝑥 or any other direction in general: 𝜌𝜌(𝜉𝜉) =

〈(ℎ(𝑥𝑥 + 𝜉𝜉) − 〈ℎ(𝑥𝑥)〉(ℎ(𝑥𝑥) − 〈ℎ(𝑥𝑥)〉〉 𝜎𝜎 2

(1.5)

as a value of 𝜉𝜉 for which function drops to 𝜌𝜌(0)/𝑒𝑒. The simplest assumption is that autocorrelation function is the same for each direction, in which case we have isotropic surface roughness. Natural surfaces have many scales of roughness, which makes autocorrelation function to be exponential. Therefore, for snow cover surface, the most commonly used function is isotropic exponential correlation function: 𝜌𝜌(𝜉𝜉) = exp �−

|𝜉𝜉| � 𝑙𝑙𝑐𝑐

(1.6)

Beside these basic physical parameters, it can be significant to mention liquid water distribution, which is very difficult to exactly estimate, but can be very important for wet snow characterization.

I.2 Active microwave remote sensing There are several remote sensing basic classifications. -

-

-

Depending whether the sensors are located on aircraft or satellite platform, it can be: o Airborne remote sensing o Spaceborne remote sensing Depending on part of the electromagnetic spectrum being used: o Optical remote sensing (ultraviolet, visible and reflected infrared spectrum parts) o Infrared remote sensing (thermal infrared spectrum part) o Microwave remote sensing (part of interest for remote sensing, 𝜆𝜆 = 1𝑚𝑚𝑚𝑚 − 1𝑚𝑚) Whether platforms are just receiving or also transmitting energy, system can be: o Passive o Active

Concerning the snow monitoring application, which is the topic of our project, we are using active microwave remote sensing with active imaging remote sensor - Synthetic Aperture Radar (SAR), which is placed on a satellite platform. The main advantages of this remote sensing type are day or night and all-whether imaging capabilities. Former due to the fact that active remote sensor provides its own source of illumination and the latter one as a consequence of EM waves low atmospheric scattering in the microwave part of the spectrum.

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I.2.1. Synthetic Aperture Radar (SAR)

RADAR (RAdio Detection And Ranging) uses the backscattered signal to discriminate between different targets and the time delay between transmitted and received signal to estimate target distance or range. Radar geometry is presented in a figure 1.1.

Platform with antennas is travelling along the A direction with a nadir (B) beneath it, while the microwave radiation illuminates swath region (C). Direction D and E are called, respectively, across and along track dimensions. Target range is determined in the first direction and target azimuth in the second one. The direct distance between the antenna and the target is called slant range.

Figure 1.1: Radar geometry The most important radar characteristics are the radiometry and the spatial resolution. Radiometric resolution is being referred to the smallest change in the intensity level that can be detected by the receiving antenna. In a digital image, it is limited by the number of quantization levels used to digitize the value of the received power. Spatial resolution refers to the size of the smallest object which can be detected. It is given as a product of range (across-track) and azimuth (along-track) resolution. Range resolution is a function of the electromagnetic pulse length and the target range (Figure 1.2a). It can be improved by reducing pulse time and thus its length. On the other side, azimuth resolution is a function of the antenna beamwidth and the target range (Figure 1.2b).

Figure 1.2: (a) range resolution, (b) azimuth resolution

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Generally, antenna beamwidth is inversely proportional to antenna length or antenna aperture surface, but significant improvement can not be made in this direction because of the platform limitations. Therefore, improved azimuth resolution is archived by doing postprocessing and by that simulating long antenna (Figure 1.3). This way, we are getting the synthetic aperture radar (SAR) image.

The target backscattering pulses are being recorded from the moment the target enters the antenna footprint until the moment when it leaves it, because of the antenna movement. The length passed by the platform during that time is the equivalent synthetic antenna length.

Figure 1.3: Synthetic antenna Radar antennas are mostly able to both transmit and receive horizontally and vertically polarized waves. Depending on which polarization combination is used, we can differ: HH, VV, HV and VH – polarized image. Data given by different polarization channels can be complementary and therefore can provide more detailed information about the target area. In order to be able to distinguish targets it is necessary to describe them in sense of measured backscattered power. Therefore a physical quantity called the backscattering coefficient is defined. The backscattering coefficient 𝜎𝜎 0 (sigma nought) is a dimensionless variable, which represents a ratio between the backscattering cross section to physical area in monostatic radar equation (1.7). It is used for distributed targets characterization. 𝑃𝑃𝑟𝑟 =

𝜆𝜆2 𝐺𝐺 2 𝑃𝑃𝑡𝑡 0 𝜎𝜎 𝐴𝐴 (4𝜋𝜋)3 𝜂𝜂𝑅𝑅4

(1.7)

where 𝑃𝑃𝑟𝑟 and 𝑃𝑃𝑡𝑡 represent, respectively, received and transmitted power, 𝐺𝐺 is antenna gain, 𝜂𝜂 efficiency, 𝜆𝜆 - wavelength, 𝑅𝑅 - distance between antenna and the target and 𝐴𝐴 - physical area. Backscattering cross section represents the area intercepting the amount of power which, when scattered isotropically, produces at the receiver a density which is equal to that backscattered by the actual target. It can be also defined simply as a ratio of total backscattered power density and power density of totally reradiated incident power: 𝜎𝜎 0 =

〈|𝐸𝐸𝑠𝑠 |2 〉 𝑃𝑃𝑖𝑖 4𝜋𝜋𝑅𝑅2

(1.8)

where nominator represents averaged scattered power density in the direction of interest, while denominator gives perfectly reradiated incident power density. In case of using several polarization channels (polarimetric model), the target is described by the scattering matrix, which forms varies depending on propagation convention (BSA or FSA), type of 12


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scattering (monostatic or bistatic) and whether the radiation is fully polarized or not (Appendix I.2). Therefore, we have: Jones matrix (FSA, fully polarized radiation), Sinclair matrix (BSA, fully polarized radiation), Mueller matrix (FSA, partly polarized radiation), Kennaugh (BSA, partly polarized radiation).

I.3 Radiative transfer theory Radiative transfer theory is based on the integro-differential equation called radiative transfer equation, which describes propagation of EM waves inside a particular medium. While propagating through some medium, EM wave have a certain energy losses. Those losses can be a consequence of the absorption or the scattering processes. EM wave energy can also be increased due to the medium emission or due to the scattered energy of other waves propagating through the medium. Mathematical description of those processes is provided by radiative transfer equation:

Figure 1.4: EM propagation through a medium 𝑑𝑑𝑑𝑑(𝑟𝑟⃗, 𝑠𝑠⃗) = −𝜅𝜅𝑎𝑎 𝐼𝐼(𝑟𝑟⃗, 𝑠𝑠⃗) − 𝜅𝜅𝑠𝑠 𝐼𝐼(𝑟𝑟⃗, 𝑠𝑠⃗) + � 𝑃𝑃(𝑟𝑟⃗, 𝑠𝑠⃗′ )𝐼𝐼(𝑟𝑟⃗, 𝑠𝑠⃗′ )𝑑𝑑Ω′ 𝑑𝑑𝑑𝑑 4𝜋𝜋

(1.9)

𝐼𝐼(𝑟𝑟⃗, 𝑠𝑠⃗) represents EM wave intensity along 𝑠𝑠⃗ direction, 𝜅𝜅𝑎𝑎 is absorption coefficient, 𝜅𝜅𝑠𝑠 – scattering coefficient, while 𝑃𝑃(𝑟𝑟⃗, 𝑠𝑠⃗ ′ ) represents phase function. On the left side of the equation (1.9) we have the change of radiation intensity along the propagation direction. First term on the right side represents radiation energy loss due to the absorption effect, second one is loss due to the scattering, while the third one represents radiation coming from 𝑠𝑠⃗′ direction which is scattered into 𝑠𝑠⃗ direction. In a more general approach, radiative transfer equation can be expanded with the positive term on 𝐾𝐾𝐾𝐾 the right side 𝜅𝜅𝑎𝑎 2 , which represents thermal emission term at microwave frequencies, with 𝐾𝐾 as 𝜆𝜆 the Boltzmann constant, 𝑇𝑇 as a temperature and 𝜆𝜆 as a wavelength.

If intensity is a scalar, then equation 1.7 is called scalar radiative transfer equation. By solving scalar equation we are getting information about radiation propagation just for one wave polarization component. On the other side, if 𝐼𝐼(𝑟𝑟⃗, 𝑠𝑠⃗) takes the form of a modified Stokes vector (1.10), solution of the appropriate vector radiative transfer equation, can be used to establish polarimetric model, which gives us information about medium scattering properties for both of the polarization components.

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〈|𝐸𝐸𝑥𝑥 |2 〉 2

⎛ ⎞ 〈�𝐸𝐸𝑦𝑦 � 〉 𝐼𝐼 = ⎜ ∗ 2〈𝑅𝑅𝑅𝑅(𝐸𝐸𝑥𝑥 𝐸𝐸𝑦𝑦 )〉 ⎟ ∗ ⎝−2〈𝐼𝐼𝐼𝐼(𝐸𝐸𝑥𝑥 𝐸𝐸𝑦𝑦 )〉⎠

(1.10)

This way, quantities 𝜅𝜅𝑎𝑎 , 𝜅𝜅𝑠𝑠 are having 4x4 matrix form, while phase function 𝑃𝑃(𝑟𝑟⃗, 𝑠𝑠⃗ ′ ) now represents phase matrix with a same (4x4) dimensions.

I.4 Image analysis The ultimate goal of the whole project is to derive SWE from the given SAR images. Therefore, the subject of my internship, examination of link between snow cover parameters and backscattering coefficient is just a part of a chain, even though critical one. The other important part is link between image and backscattering coefficient. This part is already been done during the last year internship [9] so I will use this chapter in order to present some basic information concerning this problem and also to introduce data which are going to be used for the general validation in the future after we get supplied with 'in situ' measurements data by the EDF company. Used images are results of radar acquisitions by German satellite TerraSAR-X over French Alps done on 8. February, 19. February and 2. March in 2009. They are covering area of 15 km2 of Grandes Rousses massive. All of them are done in X band, using two polarization channels, HH and VV. One of them, the result of 8. February acquisition is given in the figure 1.5. Before presenting image analysis results, it is necessary to shortly introduce the approach adopted for target modeling in the context of image analysis. According to SIRV (Spherically Invariant Random Vectors) target can be described via target vector, as: 𝑘𝑘�⃗ = √𝜏𝜏 𝑧𝑧⃗

(1.11)

where 𝜏𝜏 is a scalar quantity describing image power, while 𝑧𝑧⃗ is the Gaussian complex vector representing speckle which describes the variations within different polarization channels. Target vector is described using scattering matrix as: 𝑆𝑆𝐻𝐻𝐻𝐻 + 𝑆𝑆𝑉𝑉𝑉𝑉 �⃗ 𝑘𝑘 = �𝑆𝑆𝐻𝐻𝐻𝐻 − 𝑆𝑆𝑉𝑉𝑉𝑉 � 2𝑆𝑆𝑋𝑋𝑋𝑋

(1.12)

with S HH and S VV as a ratios between like polarized electric fields and S XX as a ration between cross polarized fields (S XX = S HV = S VH ). If we introduce mean power value μ, we can rewrite first equation as: 𝑘𝑘�⃗ = �𝜉𝜉 𝜇𝜇 𝑧𝑧⃗

(1.13)

with ξ as the normalized image power value. Exactly this quantity was in the centre of the last year internship. It was shown [9], using synthesized inverse simulator, that values of ξ are following Beta Prime probability distribution within one group of pixels.

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Figure 1.5: Massive des Grandes Rousses (8.2.2009.) ; TerraSAR-X amplitude color composition First the image pixels, according to covariance matrix, were divided into the 9 classes. For each of the classes, by applying inverse simulator, appropriate Beta Prime distribution parameters were calculated, along with mean power values, which mostly discriminate separate image regions, and because of that they are essential for our future validation process. Results are given in the form of table: Class (number of pixels) 1 (3993) 2 (3017) 3 (2576) 4 (1204) 5 (103872) 6 (50136) 7 (149160) 8 (292984) 9 (483018)

α β 1.1878 2.4560 1.0885 2.3066 1.2739 2.4972 1.2317 2.3735 1.1354 2.3768 1.8814 2.9255 1.4825 2.5945 1.1852 2.3459 1.9925 3.0387 Figure I.5: Table with image analysis results

μ 18,567.1948 22,018.9572 16,172.7780 16,691.1885 27,599.0250 14,070.9449 21,790.8051 34,931.3299 8,511.9011

Where α and β are representing Beta prime distribution parameters:

with B as a beta function.

𝛽𝛽′ (𝜉𝜉; 𝛼𝛼, 𝛽𝛽) =

1 𝜉𝜉 𝛼𝛼−1 (1 + 𝜉𝜉)−(𝛼𝛼+𝛽𝛽 ) 𝐵𝐵(𝛼𝛼, 𝛽𝛽)

15

(1.14)


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II Electromagnetic properties of a snow cover Backscattering coefficient or more generally, backscattering matrix, as it will be shown in this chapter, through radiative transfer equation solution, is a function of two group of parameters. First one are called sensor parameters (frequency, polarization and incidence angle) and they do not contain direct information about the snow cover properties. Second group of parameters – observed surface parameters (electromagnetic properties and surface roughness), especially electromagnetic properties, can be related directly to snow cover physical properties which are defining SWE. Therefore, this chapter is mostly treating the links between snow cover properties from I.2 and EM properties (II.1) and also links between EM properties and other properties with backscattering coefficient (II.2) – Figure 2.1.

Figure 2.1: Problem description

II.1 Dielectric properties II.1.1 Dielectric constant Dielectric properties of a material are characterized by its dielectric permittivity ε. This physical quantity shows how good a certain material attenuates the electric field or how much it can be �⃗ and electric field polarized if exposed to the field. It relates displacement field (electric induction) 𝐷𝐷 𝐸𝐸�⃗ vectors: �⃗ = 𝜀𝜀0 𝐸𝐸�⃗ + 𝑃𝑃�⃗ = 𝜀𝜀0 𝐸𝐸�⃗ + 𝜒𝜒𝜀𝜀0 𝐸𝐸�⃗ = 𝜀𝜀0 (1 + 𝜒𝜒)𝐸𝐸�⃗ = 𝜀𝜀0 𝜀𝜀𝑟𝑟 𝐸𝐸�⃗ = 𝜀𝜀𝐸𝐸�⃗ 𝐷𝐷

(2.1)

𝜀𝜀𝑟𝑟 = 𝜀𝜀𝑟𝑟 ′ + 𝑗𝑗𝜀𝜀𝑟𝑟 ′′ = 𝜀𝜀𝑟𝑟 ′ + 𝑗𝑗(𝜀𝜀𝑟𝑟𝑟𝑟 ′′ + 𝜀𝜀𝑟𝑟𝑟𝑟 ′′ )

(2.2)

𝑃𝑃�⃗ is polarization vector, 𝜒𝜒 is susceptibility, while 𝜀𝜀0 and 𝜀𝜀𝑟𝑟 represent respectively dielectric constant of vacuum (8.85 ∙ 10−12 𝐹𝐹/𝑚𝑚) and relative dielectric permittivity or simply dielectric constant. Principally, dielectric constant is a complex number: Real part (𝜀𝜀𝑟𝑟 ′ ) represents polarization capabilities while imaginary part (𝜀𝜀𝑟𝑟 ′′ ) gives us information about losses: 𝜀𝜀𝑟𝑟𝑟𝑟 ′′ - dielectric losses, which are consequence of “inertia effect”, where polarization vector can not follow changes of electric field vector, and therefore are especially present in the microwave region, 𝜀𝜀𝑟𝑟𝑟𝑟 ′′ - conducting losses, which are basically Joule’s losses or energy which is dissipated through the heating. II.1.2 Dielectric constant of a snow

Dielectric properties of a snow are mostly determined by the liquid water content, or by its wetness. It can be justified by the fact that dielectric constant of ice in the microwave region is around 3.15 17


Nikola BEŠIĆ

with negligible imaginary part, which is dependent on temperature but in the region between 1 and 20 GHz, it has an average value of 0.005, while water on the other side has a significantly larger dielectric constant, as it is shown in the following figure:

Figure 2.2: (a) – Real part of the water dielectric constant, (b) – Imaginary part of the water dielectric constant; ( Capture from [7], page 148, figures 2 and 3) Therefore, from the electromagnetic point of view, dry snow and wet snow are behaving like totally different materials. In case of a dry snow, which is consisted of ice particles and air, the imaginary part of the dielectric constant is negligible, while the real part can be expressed as a function of density 𝜌𝜌𝑑𝑑𝑑𝑑 : where density is given in 𝑔𝑔/𝑐𝑐𝑐𝑐3 .

𝜀𝜀 ′ = 1 + 1.9𝜌𝜌𝑑𝑑𝑑𝑑

(2.3)

For wet snow, the dielectric constant is not just a function of density and wetness, but also a function of the frequency. Therefore, two models were developed: -

for frequencies between 3 and 15 GHz: 𝜀𝜀 ′ = 1 + 1.83𝜌𝜌𝑑𝑑𝑑𝑑 + 0.02𝑤𝑤 1.015 + 𝜀𝜀 ′′ =

-

0.073𝑤𝑤 1.31 𝑓𝑓 2 1+( ) 9.07

𝑓𝑓 0.073 �9.07� 𝑤𝑤 1.31 𝑓𝑓 2 1+( ) 9.07

(2.4a)

(2.4b)

for frequencies between 15 and 37 GHz:

𝜀𝜀 ′ = 1 + 1.83𝜌𝜌𝑑𝑑𝑑𝑑 + (0.016 + 0.0006𝑓𝑓 − 1.2 × 10−5 𝑓𝑓 2 )𝑤𝑤 1.015 + 0.31 − 0.05𝑓𝑓 + 8.7 × 10−4 𝑓𝑓 2 +

𝜀𝜀 ′′ =

�0.057+0.0022𝑓𝑓−4.2×10 −5 𝑓𝑓 2 �𝑤𝑤 1.31 𝑓𝑓 1+(9.07 )2

(0.071 − 2.8 × 10−4 𝑓𝑓 + 2.8 × 10−5 𝑓𝑓 2 ) � 𝑓𝑓 2 1+( ) 9.07 18

𝑓𝑓 1.31 9.07� 𝑤𝑤

(2.5a)

(2.5b)


Nikola BEŠIĆ

In the following figure can be seen how polarization and attenuation properties of a wet snow depend on its wetness in L, C, X and Ku band.

Figure 2.3: Wet snow dielectric constant as a function of wetness with 𝜌𝜌𝑑𝑑𝑑𝑑 = 200 𝑘𝑘𝑘𝑘/𝑚𝑚3 .

It can be seen that, with increasing frequency we are reducing the liquid water content influence on the real part of dielectric constant, which in our case can mean that for higher frequencies we have smaller changes in angle due to the air-snow boundary, and also bigger Fresnel transmission coefficient. On the other side, absorption is increasing with frequency.

II.2 Backscattering mechanism Total backscattering is consisted out of four components: 1. 2. 3. 4.

Air-snow interface backscattering component Volume backscattering component Ground backscattering component Ground – volume interaction component

Figure 2.4: Backscattering components in case of one-layer snow cover

19


Nikola BEŠIĆ

Like it is explained in chapter I.3, if we are solving a vector radiative transfer equation, the different components can be expressed in the form of Mueller matrixes (FSA convention) and thus we have the polarimetric model, while in case of a scalar radiative transfer equation, components are represented through the co-polarized backscattering coefficients, which can be also derived from the Mueller matrix. 1. Air-snow interface backscattering component This component is calculated using IEM with input data: RMS height of a snow surface, correlation length of a snow surface and snow dielectric constant. The corresponding Mueller matrix (𝑴𝑴𝑨𝑨𝑨𝑨 ) is: 𝑴𝑴𝑨𝑨𝑨𝑨 = 𝑺𝑺𝒔𝒔 (𝜇𝜇𝑖𝑖 , −𝜇𝜇𝑖𝑖 , ϕs − ϕ)

(2.6)

with 𝑺𝑺𝒔𝒔 as a snow surface backscattering matrix, 𝜇𝜇𝑖𝑖 - cosine of the incident elevation angle and ϕs − ϕ difference between incident and scattered azimuth angle. 2. Vol ume backscattering component

In polarimetric case, appropriate Mueller matrix (𝑴𝑴𝑽𝑽 ) is given as: 𝑴𝑴𝑽𝑽 = 𝑻𝑻(𝜃𝜃𝑟𝑟 , 𝜃𝜃𝑖𝑖 )

𝑷𝑷(𝜇𝜇𝑟𝑟 , −𝜇𝜇𝑟𝑟 , ϕs − ϕ) −2κe d 𝜇𝜇𝑟𝑟 �1 − exp � �� 𝑻𝑻(𝜃𝜃𝑖𝑖 , 𝜃𝜃𝑟𝑟 ) 2κe 𝜇𝜇𝑟𝑟

(2.7a)

where 𝑻𝑻 is the Fresnel power transmission coefficient matrix (4x4 dimensions) for air-snow interface, 𝑷𝑷 – Rayleigh phase matrix (4x4), κe – extinction coefficient, 𝑎𝑎 – albedo, d – depth, while 𝜇𝜇𝑖𝑖 and 𝜇𝜇𝑟𝑟 are, respectively, cosines of an incident and refracted angle. The co-polarized backscattering coefficient is: σvpp = 𝑎𝑎

𝜇𝜇𝑖𝑖 2 −2κe d 𝑇𝑇𝑝𝑝𝑝𝑝 (𝜃𝜃𝑟𝑟 , 𝜃𝜃𝑖𝑖 )𝑃𝑃𝑝𝑝𝑝𝑝 (𝜇𝜇𝑟𝑟 , −𝜇𝜇𝑟𝑟 , ϕs − ϕ) �1 − exp � �� 2 𝜇𝜇𝑟𝑟

(2.7b)

with pp which can symbolize vv or hh co-polarization channel. Therefore 𝑇𝑇𝑝𝑝𝑝𝑝 and 𝑃𝑃𝑝𝑝𝑝𝑝 are the corresponding Fresnel power transmission coefficient (Appendix I.1) and the Rayleigh phase function (2.8). 𝑃𝑃𝑣𝑣𝑣𝑣 (𝜇𝜇𝑟𝑟 , −𝜇𝜇𝑟𝑟 , 𝜙𝜙𝑠𝑠 − 𝜙𝜙) = 0.75{2(1 − 𝜇𝜇𝑟𝑟2 )2 + 𝜇𝜇𝑟𝑟4 [1 + cos 2(𝜙𝜙𝑠𝑠 − 𝜙𝜙)] + 4𝜇𝜇𝑟𝑟2 (1 − 𝜇𝜇𝑟𝑟2 )𝑐𝑐𝑐𝑐𝑐𝑐(𝜙𝜙𝑠𝑠 − 𝜙𝜙)} 𝑃𝑃ℎℎ (𝜇𝜇𝑟𝑟 , −𝜇𝜇𝑟𝑟 , 𝜙𝜙𝑠𝑠 − 𝜙𝜙) = 0.75[1 + cos 2(𝜙𝜙𝑠𝑠 − 𝜙𝜙)]

(2.8a) (2.8b)

3. Ground backscattering component This component represents ground backscattering attenuated by the snow volume and by the airsnow boundary. The appropriate Mueller matrix is given as: 𝑴𝑴𝑮𝑮 = 𝑻𝑻(𝜃𝜃𝑟𝑟 , 𝜃𝜃𝑖𝑖 )𝑺𝑺𝒈𝒈 (𝜇𝜇𝑟𝑟 , −𝜇𝜇𝑟𝑟 , ϕs − ϕ)𝑻𝑻(𝜃𝜃𝑖𝑖 , 𝜃𝜃𝑟𝑟 ) exp �

−2κe d � 𝜇𝜇𝑟𝑟

where 𝑺𝑺𝒈𝒈 represents the ground surface backscattering matrix. In this case the co-polarized backscattering coefficient is given as: 𝜇𝜇𝑖𝑖 2 −2κe d g (𝜃𝜃𝑟𝑟 , 𝜃𝜃𝑖𝑖 )𝜎𝜎𝑔𝑔 (𝜃𝜃𝑟𝑟 , 𝜃𝜃𝑟𝑟 , ϕs − ϕ) exp � σpp = 𝑇𝑇𝑝𝑝𝑝𝑝 � 𝜇𝜇𝑟𝑟 𝜇𝜇𝑟𝑟

with 𝜎𝜎𝑔𝑔 – ground backscattering coefficient, which is being calculated using IEM. 20

(2.9a)

(2.9b)


Nikola BEŠIĆ

4. Ground – volume interaction component Interaction component is, in the simplest approach, consisted out of two sub-components, presented in the figure 2.4. The first one gives information about the energy which is, after volume scattering reflected by the ground in the direction of interest, while the second one represents the energy reflected in a specular direction, which is being re-directed toward the receiver by the snow volume. The Mueller matrix is given as: 1 −2κe d 𝑻𝑻(𝜃𝜃𝑟𝑟 , 𝜃𝜃𝑖𝑖 )𝑻𝑻(𝜃𝜃𝑖𝑖 , 𝜃𝜃𝑟𝑟 ) exp � � 𝑹𝑹(𝜃𝜃𝑟𝑟 , 𝜃𝜃𝑟𝑟 )𝑷𝑷(−𝜇𝜇𝑟𝑟 , −𝜇𝜇𝑟𝑟 , ϕs 𝜇𝜇𝑖𝑖 𝜇𝜇𝑟𝑟 1 − exp(−k 2 d) − ϕ) k2 1 −2κe d = 𝑻𝑻(𝜃𝜃𝑟𝑟 , 𝜃𝜃𝑖𝑖 )𝑻𝑻(𝜃𝜃𝑖𝑖 , 𝜃𝜃𝑟𝑟 ) exp � � 𝑹𝑹(𝜃𝜃𝑟𝑟 , 𝜃𝜃𝑟𝑟 )𝑷𝑷(𝜇𝜇𝑟𝑟 , 𝜇𝜇𝑟𝑟 , ϕs 𝜇𝜇𝑖𝑖 𝜇𝜇𝑟𝑟 exp(−k 2 d) − 1 − ϕ) k2

𝑴𝑴𝑮𝑮𝑮𝑮 𝟏𝟏 =

(2.10a)

𝑴𝑴𝑮𝑮𝑮𝑮 𝟐𝟐

(2.10b)

𝑹𝑹(𝜃𝜃𝑟𝑟 , 𝜃𝜃𝑟𝑟 ) is the Fresnel power reflection matrix, modified by the roughness effect (multiplied by 1 1 appropriate factor) and k 2 is κe � ′ − ′′ � where μ′ and μ′′ are volume scattering incident and μ

μ

scattered angle. In this case, both of them are 𝜇𝜇𝑟𝑟 .

Co-polarized backscattering coefficient is given as: gv

2 (𝜃𝜃 σpp 1 = 𝜇𝜇𝑖𝑖 𝑎𝑎𝑇𝑇𝑝𝑝𝑝𝑝 𝑟𝑟 , 𝜃𝜃𝑖𝑖 )𝑅𝑅𝑝𝑝𝑝𝑝 (𝜃𝜃𝑟𝑟 , 𝜃𝜃𝑟𝑟 ) gv

− ϕ)

κe d −2κe d exp � − 4k 2 σ2RMS μ2r � 𝑃𝑃(−𝜇𝜇𝑟𝑟 , −𝜇𝜇𝑟𝑟 , ϕs 𝜇𝜇𝑟𝑟 𝜇𝜇𝑟𝑟

2 (𝜃𝜃 σpp 2 = 𝜇𝜇𝑖𝑖 𝑎𝑎𝑇𝑇𝑝𝑝𝑝𝑝 𝑟𝑟 , 𝜃𝜃𝑖𝑖 )𝑅𝑅𝑝𝑝𝑝𝑝 (𝜃𝜃𝑟𝑟 , 𝜃𝜃𝑟𝑟 )

− ϕ)

κe d −2κe d exp � − 4k 2 σ2RMS μ2r � 𝑃𝑃(𝜇𝜇𝑟𝑟 , 𝜇𝜇𝑟𝑟 , ϕs 𝜇𝜇𝑟𝑟 𝜇𝜇𝑟𝑟

(2.11a)

(2.11b)

with 𝑅𝑅𝑝𝑝𝑝𝑝 (𝜃𝜃𝑟𝑟 , 𝜃𝜃𝑟𝑟 ) – the Fresnel reflection coefficient and 4k 2 σ2RMS μ2r – the roughness effect factor, k represents wavenumber in a snow medium and σRMS is the RMS height of a ground. Generally, there are three more types of interaction, which can be taken into the account using a more sophisticated model (Figure 2.5).

Figure 2.5: Additional ground – volume interaction backscattering sub-components 21


Nikola BEŠIĆ

Case A describes the situation when we have double ground specular reflection. After being reflected by the ground first time, the wave is being backscattered downwards by snow volume and then reflected in the direction of interest by a second ground reflection. The corresponding Mueller matrix is: 𝑴𝑴𝑮𝑮𝑮𝑮 𝐀𝐀 =

1 −2κe d 𝟐𝟐 𝑻𝑻(𝜃𝜃𝑟𝑟 , 𝜃𝜃𝑖𝑖 )𝑻𝑻(𝜃𝜃𝑖𝑖 , 𝜃𝜃𝑟𝑟 ) exp � � 𝑹𝑹 (𝜃𝜃𝑟𝑟 , 𝜃𝜃𝑟𝑟 )𝑷𝑷(−𝜇𝜇𝑟𝑟 , 𝜇𝜇𝑟𝑟 , ϕs 𝜇𝜇𝑖𝑖 𝜇𝜇𝑟𝑟 −κ d 1 − exp � 𝜇𝜇 e � 𝑟𝑟 − ϕ) 2κe

(2.12)

In the case B wave is being scattered by volume at some arbitrary angle and then backscattered by the ground toward the receiver. This sub-component is represented as: 𝑴𝑴

𝑮𝑮𝑮𝑮

𝐁𝐁

1 1 exp(−𝑘𝑘1 𝑑𝑑) 2𝜋𝜋 exp(𝑘𝑘2 𝑑𝑑) − 1 = 𝑻𝑻(𝜃𝜃𝑟𝑟 , 𝜃𝜃𝑖𝑖 )𝑻𝑻(𝜃𝜃𝑖𝑖 , 𝜃𝜃𝑟𝑟 ) � � 𝑆𝑆𝑅𝑅12 (𝜇𝜇𝑟𝑟 , 𝜋𝜋 𝜇𝜇𝑖𝑖 𝜇𝜇′ 𝑘𝑘2 0 0 + 𝜙𝜙𝑖𝑖 ; −𝜇𝜇′ , 𝜙𝜙 ′ )𝑃𝑃(−𝜇𝜇′ , 𝜙𝜙 ′ ; −𝜇𝜇𝑟𝑟 , 𝜙𝜙𝑖𝑖 ) 𝑑𝑑𝜙𝜙 ′ 𝑑𝑑𝑑𝑑′ 1

with 𝑘𝑘1 defined as the κe � ′ + μ

1

μ ′′

(2.13)

�.

Finally, we have the wave scattered by the ground at arbitrary angle and then re-directed by the snow volume. The Mueller matrix in this case is: 1 1 exp(−𝑘𝑘1 𝑑𝑑) 2𝜋𝜋 1 − exp(−𝑘𝑘2 𝑑𝑑) 𝑻𝑻(𝜃𝜃𝑟𝑟 , 𝜃𝜃𝑖𝑖 )𝑻𝑻(𝜃𝜃𝑖𝑖 , 𝜃𝜃𝑟𝑟 ) � � 𝑃𝑃(𝜇𝜇𝑟𝑟 , 𝜋𝜋 𝜇𝜇𝑖𝑖 𝜇𝜇′ 𝑘𝑘2 0 0 + 𝜙𝜙𝑖𝑖 ; 𝜇𝜇′ , 𝜙𝜙 ′ ) 𝑆𝑆𝑅𝑅12 (𝜇𝜇′ , 𝜙𝜙 ′ ; −𝜇𝜇𝑟𝑟 , 𝜙𝜙𝑖𝑖 ) 𝑑𝑑𝜙𝜙 ′ 𝑑𝑑𝑑𝑑′

𝑴𝑴𝑮𝑮𝑮𝑮 𝐂𝐂 =

22

(2.14)


Nikola BEŠIĆ

III Dry snow analysis As it is stated in the introduction, dry snow is snow at the temperatures below 0 oC, which doesn’t contain any liquid water between ice particles. This chapter covers dry snow analysis starting with a brief theoretical description of dry snow medium modeling. Further, the core of the report – snow backscattering sensitivity analysis done using the previously presented models and the backscattering mechanism from the Chapter II, are given, along with physical explanations, justifications and conclusions. At the end, the semi-empirical model developed by American scientists Shi and Dozier [1,2], built on the assumptions which were proven during sensitivity analysis, is described in details.

III.1 Theoretical introduction III.1.1 Rayleigh scattering model The elastic scattering case when the particle size is much smaller than the wavelength is called Rayleigh scattering and is being described using Rayleigh scattering model. It assumes that, when the electromagnetic wave hits the particle, an oscillatory dipole is induced inside it. That dipole reradiates part of the induced energy which is recognized as a scattered energy. In the simplest case we can suppose the spherical shape of the particle. Intensity of the filed induced in the spherical ice particle (𝐸𝐸�⃗𝑖𝑖𝑖𝑖𝑖𝑖 ) as a function of the incident field (𝐸𝐸�⃗𝑖𝑖 ) and dielectric constants of a particle and a medium (𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 and 𝜀𝜀0 ) is given by: 𝐸𝐸�⃗𝑖𝑖𝑖𝑖𝑖𝑖 =

3𝜀𝜀0 𝐸𝐸�⃗ 𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 + 𝜀𝜀0 𝑖𝑖

(3.1)

Not all the energy is being reradiated, one part is absorbed by the particle which in this case means transferred into the heat energy. According to Ohm’s law, power absorbed by the particle is defined as: 1 ′′ 𝑃𝑃𝑎𝑎 = 𝜔𝜔𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 𝜐𝜐0 �𝐸𝐸�⃗𝑖𝑖𝑖𝑖𝑖𝑖 � 2

(3.2)

′′ where 𝜔𝜔 represents angular frequency, 𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 is imaginary part of the particle dielectric constant and 𝜐𝜐0 is volume of the particle. Knowing total power absorbed by the particle, we can define a quantity called absorption cross section, which represents the surface intercepting part of the incident energy which will be absorbed: ′′ 𝑃𝑃𝑎𝑎 𝑃𝑃𝑎𝑎 𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 3𝜀𝜀0 2 𝜎𝜎𝑎𝑎 = = = 𝑘𝑘 𝜐𝜐 � � ��⃗𝑖𝑖 � |𝐸𝐸𝑖𝑖 |2 𝜀𝜀0 0 𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 + 𝜀𝜀0 �𝑆𝑆 2𝜂𝜂

(3.3)

��⃗𝑖𝑖 � is the intensity of the active incident power density (Poynting’s vector real part module), 𝑘𝑘 is �𝑆𝑆 the wavenumber, while 𝜂𝜂 represents medium wave impedance of a free space. In order to describe analytically the scattering effect, it is convenient to represent both incident and scattered field through the sums of their orthogonal (vertical and horizontal) components.

23


Nikola BEŠIĆ

�⃗𝑖𝑖 ��⃗ 𝐸𝐸𝑖𝑖 = 𝐸𝐸𝑣𝑣𝑣𝑣 𝑣𝑣⃗𝑖𝑖 + 𝐸𝐸ℎ𝑖𝑖 ℎ ��⃗𝑠𝑠 = 𝐸𝐸𝑣𝑣𝑣𝑣 𝑣𝑣⃗𝑠𝑠 + 𝐸𝐸ℎ𝑠𝑠 ℎ�⃗𝑠𝑠 𝐸𝐸

(3.4)

��⃗𝑠𝑠 , ��⃗ Now, it is possible to relate them using the scattering amplitude (𝑓𝑓(𝑘𝑘 𝑘𝑘𝑖𝑖 )) matrix (3.6). This matrix is derived by calculating the induced dipole moment (𝑝𝑝⃗) and then expressing scaterred field in terms of this quantity:

� where:

𝑝𝑝⃗ = 𝜐𝜐0 (𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 − 𝜀𝜀0 )𝐸𝐸�⃗𝑖𝑖𝑖𝑖𝑖𝑖 𝑘𝑘 2 ��⃗ ��⃗𝑠𝑠 × 𝑝𝑝⃗) ��⃗ 𝐸𝐸𝑠𝑠 = − 𝑘𝑘 × (𝑘𝑘 4𝜋𝜋𝜀𝜀0 𝑠𝑠

𝑓𝑓 𝐸𝐸𝑣𝑣𝑣𝑣 � = � 𝑣𝑣𝑣𝑣 𝐸𝐸ℎ𝑠𝑠 𝑓𝑓ℎ𝑣𝑣

𝑓𝑓𝑣𝑣ℎ 𝐸𝐸𝑣𝑣𝑣𝑣 𝑣𝑣⃗ 𝑣𝑣⃗ � � �=𝑓𝑓 � 𝑠𝑠 𝑖𝑖 𝑓𝑓ℎℎ 𝐸𝐸ℎ𝑖𝑖 0 ℎ�⃗ 𝑣𝑣⃗

𝑠𝑠 𝑖𝑖

3 𝑓𝑓0 = 𝑘𝑘 2 𝑟𝑟𝑖𝑖𝑖𝑖𝑖𝑖

𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 − 𝜀𝜀0 𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 + 2𝜀𝜀0

�⃗𝑖𝑖 𝐸𝐸𝑣𝑣𝑣𝑣 𝑣𝑣⃗𝑠𝑠 ℎ �� ⃗𝑠𝑠 ℎ �⃗𝑖𝑖 𝐸𝐸ℎ𝑖𝑖 ℎ

(3.5a) (3.5b)

(3.6)

(3.7)

Scattering cross section of the particle, defined as a surface intercepting the amount of the incident power which will be scattered in some direction is being calculated by integrating the squared module of the scattering amplitude over the corresponding solid angle: 2

��⃗𝑠𝑠 , ��⃗ 𝜎𝜎𝑠𝑠 = ��𝑓𝑓(𝑘𝑘 𝑘𝑘𝑖𝑖 )� 𝑑𝑑𝛺𝛺𝑠𝑠 =

8𝜋𝜋 𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 − 𝜀𝜀0 2 3 |𝑓𝑓0 |2 = 2𝜐𝜐0 𝑘𝑘 4 𝑟𝑟𝑖𝑖𝑖𝑖𝑖𝑖 � � 3 𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 + 2𝜀𝜀0

(3.8)

In the Rayleigh scattering model, we are assuming independent scattering (Figure 3.1). Wave scattered from the particle is not affecting any other particle as an incident wave. It basically means that the total backscattered field can be expressed as a sum of each particle backscattered field.

Figure 3.1: Independent backscattering In this case, absorption coefficient, parameter used in Radiative Transfer Equation (I.3) is derived by multiplying the absorption cross section of the single particle with the number of particles per volume: ′′ 𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 3𝜀𝜀0 2 𝜅𝜅𝑎𝑎 = 𝑛𝑛𝑖𝑖𝑖𝑖𝑖𝑖 𝜎𝜎𝑎𝑎 = 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖 𝑘𝑘 � � 𝜀𝜀0 𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 + 𝜀𝜀0

24

(3.9)


Nikola BEŠIĆ

The same approach is used for the scattering coefficient: 𝜅𝜅𝑠𝑠 = 𝑛𝑛𝑖𝑖𝑖𝑖𝑖𝑖 𝜎𝜎𝑠𝑠 =

If we define extinction cross section as:

3 2𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖 𝑘𝑘 4 𝑟𝑟𝑖𝑖𝑖𝑖𝑖𝑖

𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 − 𝜀𝜀0 2 � � 𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 + 2𝜀𝜀0

(3.10)

𝜎𝜎𝑒𝑒 = 𝜎𝜎𝑎𝑎 + 𝜎𝜎𝑠𝑠

(3.11)

𝜅𝜅𝑒𝑒 = 𝑛𝑛𝑖𝑖𝑖𝑖𝑖𝑖 𝜎𝜎𝑒𝑒 = 𝜅𝜅𝑎𝑎 + 𝜅𝜅𝑠𝑠

(3.12)

The extinction coefficient is given as:

In case of a Vector Radiative Transfer Equation, when intensity is the Stokes vector, the extinction coefficient is a matrix, derived by multiplying a unity matrix with a given coefficient. It is important to notice that this kind of diagonal matrix is valid only for spherical particles. For independent scattering, the phase matrix is given as: 2

2 �⃗𝑖𝑖 � �⃗𝑖𝑖 �(𝑣𝑣⃗𝑠𝑠 𝑣𝑣⃗𝑖𝑖 ) �𝑣𝑣⃗𝑠𝑠 ℎ �𝑣𝑣⃗𝑠𝑠 ℎ 0 ⎡ (𝑣𝑣⃗𝑠𝑠 𝑣𝑣⃗𝑖𝑖 ) ⎤ 2 2 ⎢ �⃗ ⎥ �⃗ �⃗ �⃗ �ℎ 𝑣𝑣 ⃗ ��ℎ ℎ � 0 �⃗ �⃗ 𝑠𝑠 𝑖𝑖 𝑠𝑠 𝑖𝑖 �ℎ 𝑣𝑣 ⃗ � �ℎ ℎ � 𝑠𝑠 𝑖𝑖 𝑠𝑠 𝑖𝑖 ⎥ 𝑃𝑃 = 𝑛𝑛𝑖𝑖𝑖𝑖𝑖𝑖 |𝑓𝑓0 |2 ⎢ ⎢ �⃗ ⎥ � ⃗ �⃗ �⃗ �⃗ (𝑣𝑣 )�ℎ ⃗ 𝑣𝑣 ⃗ ℎ � + �𝑣𝑣 ⃗ ℎ ��ℎ 𝑣𝑣 ⃗ � 0 �⃗ �⃗ �⃗ 𝑠𝑠 𝑖𝑖 𝑠𝑠 𝑖𝑖 𝑠𝑠 𝑖𝑖 𝑠𝑠 𝑖𝑖 ⎢2�ℎ𝑠𝑠 𝑣𝑣⃗𝑖𝑖 �(𝑣𝑣⃗𝑠𝑠 𝑣𝑣⃗𝑖𝑖 ) �𝑣𝑣⃗𝑠𝑠 ℎ𝑖𝑖 ��ℎ𝑠𝑠 ℎ𝑖𝑖 � ⎥ �⃗𝑠𝑠 ℎ �⃗𝑖𝑖 � − �𝑣𝑣⃗𝑠𝑠 ℎ �⃗𝑖𝑖 ��ℎ �⃗𝑠𝑠 𝑣𝑣⃗𝑖𝑖 �⎦ (𝑣𝑣⃗𝑠𝑠 𝑣𝑣⃗𝑖𝑖 )�ℎ 0 0 0 ⎣

(3.13)

Using equations 3.7 and 3.9 it can be shown that phase matrix is a product of a scattering coefficient and a polarization phase matrix: 2

2 �⃗𝑖𝑖 � �𝑣𝑣⃗𝑠𝑠 ℎ ⎡ (𝑣𝑣⃗𝑠𝑠 𝑣𝑣⃗𝑖𝑖 ) 2 2 ⎢ 3 �⃗𝑠𝑠 𝑣𝑣⃗𝑖𝑖 � �⃗𝑠𝑠 ℎ �⃗𝑖𝑖 � �ℎ �ℎ 𝑃𝑃 = 𝜅𝜅𝑠𝑠 ⎢ 8𝜋𝜋 ⎢ �⃗ �⃗ �⃗ �⃗ ⎢2�ℎ𝑠𝑠 𝑣𝑣⃗𝑖𝑖 �(𝑣𝑣⃗𝑠𝑠 𝑣𝑣⃗𝑖𝑖 ) �𝑣𝑣⃗𝑠𝑠 ℎ𝑖𝑖 ��ℎ𝑠𝑠 ℎ𝑖𝑖 � 0 0 ⎣

�⃗𝑖𝑖 �(𝑣𝑣⃗𝑠𝑠 𝑣𝑣⃗𝑖𝑖 ) �𝑣𝑣⃗𝑠𝑠 ℎ �⃗𝑠𝑠 𝑣𝑣⃗𝑖𝑖 ��ℎ �⃗𝑠𝑠 ℎ �⃗𝑖𝑖 � �ℎ

0

⎤ ⎥ ⎥ ⎥ �⃗𝑠𝑠 ℎ �⃗𝑖𝑖 � + �𝑣𝑣⃗𝑠𝑠 ℎ �⃗𝑖𝑖 ��ℎ �⃗𝑠𝑠 𝑣𝑣⃗𝑖𝑖 � (𝑣𝑣⃗𝑠𝑠 𝑣𝑣⃗𝑖𝑖 )�ℎ 0 ⎥ �⃗𝑠𝑠 ℎ �⃗𝑖𝑖 � − �𝑣𝑣⃗𝑠𝑠 ℎ �⃗𝑖𝑖 ��ℎ �⃗𝑠𝑠 𝑣𝑣⃗𝑖𝑖 �⎦ (𝑣𝑣⃗𝑠𝑠 𝑣𝑣⃗𝑖𝑖 )�ℎ 0 0

(3.14)

The main limitation of this theory is the condition required for independent scattering, namely that particles must be separated by a distance which is significantly larger than their size. In other words saying, each particle must be in the far field of other particles. Therefore, it is applicable only for low-dense dry snow, where we can assume existence of ice particles only (hosted by air). III.1.2 QCA model In case of a dense media, when we have that distances between particles are quite small, independent scattering assumption is not valid anymore. It is necessary to take multiple scattering effect into account (Figure 3.2). Like it is shown in the figure below, the field scattered in a direction, which is not corresponding to a receiver direction, is behaving like an incident field in case of other particles. This way, we have that the backscattered wave is not just a consequence of the incident wave coming directly from a transmitter, like it is the case with independent scattering, but also a consequence of other particles scattered fields. Therefore, multiple scattering is dependent scattering and thus also a function of particles distribution.

25


Nikola BEŠIĆ

Figure 3.2: Multiple scattering Quasi Crystalline Approximation (QCA) is approximation used in solving Foldy-Lax multiple scattering equations in order to get averaged Green’s functions (Appendix I.3). Having Green’s functions, it is possible to derive effective dielectric permittivity of the medium, which introduces influence of multiple scattering on radiative transfer equation parameters and therefore on backscattering coefficient. First it is necessary to define depolarization factors, which are introducing particle spheroid shape: 𝐿𝐿𝑐𝑐 = − 𝑎𝑎 2

(1 − 𝑒𝑒̃ 2 ) 1 1 − 𝑒𝑒̃ �1 + ln � �� 2 𝑒𝑒̃ 2𝑒𝑒̃ 1 + 𝑒𝑒̃ 1 𝐿𝐿𝑎𝑎 = (1 − 𝐿𝐿𝑐𝑐 ) 2

(3.15a) (3.15b)

where 𝑒𝑒̃ = �1 − � � represents eccentricity, with 𝑎𝑎 and 𝑐𝑐 as the lengths of the spheroid semi-axes. 𝑐𝑐

1

For a special case, considered in our simulations, when we have a sphere, 𝐿𝐿𝑐𝑐 = 𝐿𝐿𝑎𝑎 = .

Generally, effective dielectric constant of the medium is given as a tensor: 𝑒𝑒𝑒𝑒𝑒𝑒

𝑒𝑒𝑒𝑒𝑒𝑒


with 𝜀𝜀𝑎𝑎 and 𝜀𝜀𝑐𝑐 directions.


𝑒𝑒𝑓𝑓𝑓𝑓

𝜀𝜀̿𝑒𝑒𝑒𝑒𝑒𝑒 = 𝜀𝜀𝑎𝑎 𝑥𝑥⃗𝑥𝑥⃗ + 𝜀𝜀𝑎𝑎 𝑦𝑦⃗𝑦𝑦⃗ + 𝜀𝜀𝑐𝑐

𝑧𝑧⃗𝑧𝑧⃗

3

(3.16)

as the scalar effective dielectric constants corresponding to appropriate spheroid

Real part of the medium effective dielectric constant, normalized to vacuum dielectric constant, is given as: 𝑒𝑒𝑒𝑒𝑒𝑒 ′

𝜀𝜀𝜇𝜇

=1+

𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖 (𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 − 1) 1 + (1 − 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖 )(𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 − 1)𝐿𝐿𝜇𝜇

(3.17a)

and the imaginary part, which contains terms related to multiple scattering effect is: 𝑒𝑒𝑒𝑒𝑒𝑒 ′′ 𝜀𝜀𝜇𝜇

𝑘𝑘 3 𝜐𝜐0 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖 (𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 − 1)2 = 𝑆𝑆 6𝜋𝜋 �1 + (1 − 𝑓𝑓 )(𝜀𝜀 − 1)𝐿𝐿 �2 0 𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖 𝜇𝜇 26

(3.17b)


Nikola BEŠIĆ

where 𝑆𝑆0 represents the structure factor (eq. 3.18), derived by applying Fourier transform on total corelation function, giving us information about the particles position in relative sense. 𝜇𝜇 can be 𝑎𝑎 or 𝑐𝑐, depending on the direction. (1 − 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖 )4 𝑆𝑆0 = (1 + 2𝑓𝑓𝑖𝑖𝑖𝑖𝑒𝑒 )2

(3.18)

In case of spherical particles, which is the considered in our simulations, the medium effective dielectric constant is the same for each direction: 3𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖 (𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 − 1) 3 + (1 − 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖 )(𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 − 1) 3 𝑘𝑘 𝜐𝜐0 3𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖 (𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 − 1)2 = 𝑆𝑆 2𝜋𝜋 [3 + (1 − 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖 )(𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 − 1)]2 0

𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 ′ = 1 +

𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 ′′

(3.19a) (3.19b)

The effective wavenumber in the medium consisted of spherical particles, can now be defined as: 𝐾𝐾 = 𝐾𝐾 ′ + 𝑗𝑗𝑗𝑗 ′′ =

2𝜋𝜋 2𝜋𝜋 𝑅𝑅𝑅𝑅 ��𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 ′ + 𝑗𝑗𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 ′′ � + 𝑗𝑗 𝐼𝐼𝐼𝐼 ��𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 ′ + 𝑗𝑗𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 ′′ � 𝜆𝜆 𝜆𝜆

(3.20)

Extinction coefficient, as a radiative transfer equation, is expressed as: 𝜅𝜅𝑒𝑒 = 2𝐾𝐾 ′′ =

4𝜋𝜋 𝐼𝐼𝐼𝐼 ��𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 ′ + 𝑗𝑗𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 ′′ � 𝜆𝜆

(3.21)

In case of a dry snow, we can roughly assume that we have ice particles placed in air host medium. As it is shown in the second chapter, imaginary part of ice dielectric constant is negligible so we can consider that there are no any absorption losses and that the whole extinction is due to the scattering effect. In the following figure (3.3), extinction coefficient as a function of ice volume fraction is presented. For multiple scattering, after a specific value of volume fraction, the extinction decreases. This is opposite to the independent Rayleigh scattering extinction, where it linearly grows with increasing volume fraction. This can be explained by the coherent scattering effect in case of a dense medium.

Figure 3.3: Extinction coefficient comparison

27


Nikola BEŠIĆ

III.2 Sensitivity analysis This section contains dry snow backscattering simulation results using both of the presented theories. Then, appropriate physical interpretations of those results are justifying the approach of Shi and Dozier used to obtain the semi-empirical model for dry snow SWE estimation, presented in III.3. Sensitivity studies for three microwave frequency bands (L, C and X) are done according to various snow cover density, which ranges from 180 kg/m3 to 220 kg/m3, and depth ranging from 0.2 m to 1 m, as it will be shown on the appropriate figures. The surface is described using the exponential isotropic correlation function with 1 cm of RMS height and 5 cm of correlation length for both of the surfaces (snow and ground). For ground dielectric constant the value 6 is taken. We were considering that one ice particle has the dielectric constant which is purely real – 3.15, while the host medium is air and therefore has the dielectric constant – 1. Ice particle is considered to be spherical and its size is calculated using the formula: 𝑟𝑟𝑖𝑖𝑖𝑖𝑖𝑖 = 𝑑𝑑 + (1 − 𝑑𝑑)(4 − 𝑠𝑠)[𝑚𝑚−4 ]

(3.22)

where 𝑑𝑑 and 𝑠𝑠 represent respectively dendricity and sphericity which can take any value between 0 and 1. In the the simulations values 0.5 are taken for both quantities. This gives radius of 0.225 mm, which is consistent with empirical data [3]. III.2.1 L-band The sensitivity studies in L-band (1-2 GHz) were done for 1.5 GHz frequency value.

Figure 3.4: Rayleigh model; (left) vv polarization, (right) hh polarization In each sub-figure, the green plot represents the backscattering as function of the incident angle for the fixed depth and various densities. The red plot represents the backscattering vs angle for fixed density and various depths. Some conclusions can be directly drawn from this (3.4 and 3.5): -

The backscattering is sensitive to the change in density, but almost independent with respect to the depth;

28


-

-

Nikola BEŠIĆ

Results in case of applying the Rayleigh model are almost identical to the ones resulting from the QAM model. The two figures are almost identical. Backscattering curves for different polarization are almost the same for smaller incident angles, but as we are approaching to bigger incident angles, the backscattering in the vv channel is getting stronger. As it can be seen in the magnified circles, the backscattering for the whole angle range is increasing when density increase. The same thing is with the increase in depth, but if we take into the account a much bigger sensitivity in case of the density, this phenomen becomes really interesting. It seems quite illogical, because larger density means bigger probability for layer scattering and absorption effect, but as it will be shown using backscattering sub-components analysis, there is quite a reasonable explanation for this behaviour. The same effect in case of the depth increase has, as it will be shown, the different cause.

Figure 3.5: QCA model; (left) vv polarization, (right) hh polarization If we recall that ice dielectric constant is purely real, as it is the case with the host medium – air, we can conclude that dry snow absorption can be neglected. On the other side, in the L band, incident EM waves have quite a big wavelength, for given frequency of 1.5 GHz the wavelength is 0.2 m. It is significantly bigger than the particle size, which is 0.225 mm in this simulation. This ratio, which is of 103 order, causes small dry snow layer scattering. Therefore, as it can be seen in the figure 3.6, volume and volume-ground components, which are having as their source ice particles scattering, are negligible.

Figure 3.6: QCA model, vv polarization; (green) Density sensitivity, (red) Depth sensitivity 29


Nikola BEŠIĆ

Air-snow component, even though significantly smaller than ground component cannot be neglected. Small value of this component can be explained by low dielectric constant of a dry snow layer, according to 2.3 formula. It is independent of depth, but significantly dependent on density, because dry snow layer dielectric constant is exclusively function of density. Absolutely dominant component is ground backscattering component, which was expected considering the fact that extinction is very small. Therefore, this component is almost independent of depth, because depth affects it through extinction phenomena. On the other side, it is quite sensitive to density, due to the dielectric constant change, which affects air-snow boundary transmission through Fresnel transmission coefficients (Appendix I.1), dry snow propagation direction (angle) through Snell refraction angle (Appendix I.1), wavelength in this medium (change of propagation medium) and snow-ground dielectric contrast. The second observation about the identical results can be justified by the fact that dry snow density is relatively small, which along with the fact that incident wavelength is significantly bigger than a spherical particle, gives perfect conditions for Rayleigh scattering model application. Generally more sophisticated, QCA model, in this case almost fits Rayleigh model, because multiple scattering effect, included through extinction, doesn’t affect backscattering significantly.

(a)

(b)

Figure 3.7: hh polarization, density sensitivity; (a) Rayleigh model, (b) QAM model

From the figure 3.7 can be seen that dominant components, ground and air-snow are identical, which is consequence of the fact that they are not affected by extinction, therefore multiple included multiple scattering doesn’t make any difference. Even though negligible, it is interesting to comment difference in volume and volume-ground components for two models. In case of a Rayleigh model, we have direct proportionality between volume backscattering and density, while in case of QCA that proportionality is inverse one. It can be explained by the figure 3.3. Given that dry snow densities correspond to ice volume fractions between 0.196 and 0.24, and for this range the Rayleigh extinction (scattering) coefficient is increased, the QCA extinction (scattering) coefficient decreases. The same way of reasoning can be applied to ground-volume interaction component, but here it is interesting to notice an increase of backscattering around the specific angle and even a change in dependence for Rayleigh model. This behavior, which is also present in a similar way in figure 3.6 can be explained using the formulas 2.12a and 2.12b. It is consequence of air-snow transmission and ground reflection dependence on angle and snow cover density. Angular dependence causes apparent curve maximum, while the decrease of their product due to the density increase causes change in backscattering dependence on density.

30


Nikola BEŠIĆ

As it is already stated, at this frequency, dry snow cover backscattering is mostly consequence of the ground backscattering. Therefore, it is mostly determined by the ground properties. In the following figure (3.8) we can see comparison between bare ground and snow-covered ground backscattering. Also, we can see comparison with dry snow ground backscattering component.

Figure 3.8: QCA model, hh and vv polarizations; Comparison with bare ground backscattering. Now the third observed fact can be explained easy. The cause of such a relation between two snow backscattering polarization channels is the relation between ground backscattering polarization channels. For bigger angles vv polarization has bigger intensity. This conclusion can be useful to choose the acquisition polarization channel. In the Figure 3.8 we can again observe and now finally explain the fact that the dominant, ground backscattering is increasing with density increase. The following figure gives density influence on ground backscattering component.

Figure 3.9: Density influence on ground backscattering parameters On the left side of the figure, the changes that are increasing ground backscattering component with density increase are presented, while on the right side are those which are decreasing the same one. Each one will be explained separately. -

𝜃𝜃𝑟𝑟 - refraction angle. According to Snell’s law: sin 𝜃𝜃𝑟𝑟 = � 31

𝜀𝜀0 sin 𝜃𝜃𝑖𝑖 𝜀𝜀𝑠𝑠

(3.23)


Nikola BEŠIĆ

with increasing 𝜀𝜀𝑠𝑠 (which is consequence of 𝜌𝜌𝑠𝑠 increase) we have decrease of refracted angle for one incident angle. If we take a look on figure 3.8, we can notice that for both copolarization channels, backscattering is bigger for smaller angles. -

𝜆𝜆𝑠𝑠 - snow layer wavelength. Considering the fact that frequency is the quantity which has absolute character, while wavelength depends on medium according to: 𝜆𝜆𝑠𝑠 =

𝑐𝑐𝑠𝑠 𝑐𝑐0 1 = 𝑓𝑓 𝑓𝑓 �𝜀𝜀𝑠𝑠

(3.24)

with increase of 𝜀𝜀𝑠𝑠 we have decrease of 𝜆𝜆𝑠𝑠 , which makes that ground appears rougher to the EM wave propagating through the snow. Therefore, backscattering is increased. -

𝑇𝑇𝑎𝑎𝑎𝑎 – Fresnel power transmission coefficient. As it can be seen from the formulas given in Appendix I.1, increased dielectric contrast between two medium reduces power transmission. That causes decrease in backscattering.

-

𝜅𝜅𝑒𝑒 – extinction coefficient. Extinction as a function of density is explained in details during the theoretical introduction and finally presented in the figure 3.3.

-

𝜀𝜀 𝑠𝑠

𝜀𝜀 𝑔𝑔

– snow-ground dielectric contrast. By decreasing dielectric contrast, we are reducing

ground reflectivity. Obviously, in L-band case, considering the fact that we have backscattering increase with density increase, first two factors are the dominant ones. Therefore, we can conclude that in L-band density is affecting backscattering mostly through the change of refracted angle and shortening of wavelength. American scientists Shi and Dozier were led exactly by this idea when they developed the semi-empirical model, which will be given and explained in the III.3 part. This model gives relation between backscattering coefficients in two co-polarized channels as a function of incident wavelength and the incident angle and suppresses dependence on dielectric properties of the ground and roughness.

(a) (b) Figure 3.10: hh polarization, depth sensitivity: (a) Rayleigh model, (b) QCA model Like it is already stated, depth sensitivity is the direct function of extinction. In the dry snow case, where we have that ground backscattering component is the dominant one, it is primary because of the fact that for higher extinction we have bigger sensitivity of attenuation term on depth. Like it can be seen in the figure 3.10, volume and volume-ground interaction components are quite

32


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sensitive to depth, so the claim can be supported also by the fact that with the higher extinction those components are giving bigger contribution to the total backscattering. In case of L-band, extinction is almost negligible so that the dominant ground backscattering component is not sensitive to depth variation, while volume and volume-ground components, can be considered as a negligible. Even though negligible, they are much more sensitive to depth variation then the first, dominant one, and therefore, those two components are causing slight increase of backscattering for depth increase. Generally, measurements done in L-band can hardly be used for depth estimation. III.2.2 C-band The sensitivity studies for C-band (4 – 8 GHz) are done for 6 GHz frequency value.

Figure 3.11: Rayleigh model; (left) vv polarization, (right) hh polarization The differences which can be notices, comparing to the L-band observations are: -

Sensitivity is increased, both to the density and to the depth change. It is particularly obvious in the case of a density variation, especially for smaller angles. Density sensitivity direction is changed, while the depth sensitivity direction remains the same. At this frequency, the density increase causes the decrease in the backscattering coefficient. It is especially interesting to notice that this behaviour is characteristic of both models. Even before the detailed analysis, this can point to the fact that extinction is not so influential again.

Figure 3.12: QCA model; (left) vv polarization, (right) hh polarization 33


Nikola BEŠIĆ

In order to justify the increased sensitivity it is necessary to analyze the behavior of backscattering components and their influence on total backscattering.

Figure 3.13: Rayleigh model, hh polarization; (left) density sensitivity, (right) depth sensitivity Even though slightly increased, volume component again can be considered as negligible. Volumeground component is even decreased, which can be explained by the fact that the increase in frequency reduces ground specular reflection, which is the source of this component. It can be noticed that volume-ground sensitivity direction is also changed relative to the L-band (figure 3.7). This phenomena is again caused by the air-snow transmission and ground reflection complex angular and density dependence. Air-snow component is increased, because frequency increase makes snow surface appearing rougher to the incident waves, but its value is, as compared to the ground backscattering component is still quite small. Therefore, the cause of increased sensitivity is increased sensitivity of the again absolutely dominant, ground backscattering component. On the other side, slightly increased depth sensitivity is justified by the increased volume component, which is significantly sensitive to depth and by increased but still quite small depth sensitivity of ground backscattering component. Both of the facts are consequence of increased extinction in C-band (figure 3.3). The fact observed primary is that the direction of depth sensitivity relative to the one in L-band is not changed as the consequence of the same volume component sensitivity direction, which is independent of frequency. Simply, always for bigger depth and thus bigger snow volume, volume backscattering component is increased.

(a) (b) Figure 3.14: Influence of separate terms on ground backscattering (hh); (a) Rayleigh, (b) QCA 34


Nikola BEŠIĆ

Trying to explain the cause of increased sensitivity and justify second mentioned observation we have analyzed dependence of particular terms affecting ground backscattering component on density and their influence on total ground backscattering: -

-

-

Fresnel power transmission at air-snow boundary and propagation angles. Like it can be concluded according to the equation given in the Appendix I.1 increase in density causes decrease in power transmission. Figure 3.14 shows that this affect can not be neglected, it has certain influence on ground backscattering component, especially at the bigger angles. Attenuation due to the extinction. Theoretical introduction gives all the necessary information concerning extinction and the difference between sensitivity direction for two models is justified by figure 3.3. Because of relatively big value of the attenuation term (inversely proportional to extinction and depth) which is consequence of the small extinction coefficient value, and its small sensitivity to density change, this term is not affecting this component value. Such a result definitely confirms the assumption from the second observation. Ground backscattering in the presence of snow. This term dominantly determines ground backscattering component. By recalling figure 3.9, after eliminating Fresnel transmission and extinction influence, we can according to figure 3.14 claim that change in sensitivity direction relative to L-band comes from the dominance of dielectric contrast sensitivity to the density change. By knowing that (figure 3.15) dielectric contrast sensitivity is not affected by the frequency (for those given, standard dry snow variation of dielectric contrast), we can conclude that change of sensitivity direction comes from the decrease of the ground backscattering sensitivity to the shortening of the wavelength and change of angle (figure 3.16). This decrease can be explained by the characteristic of the ground backscattering which is becoming more isotropic according to the angle change with frequency increase. This effect is even increased with shortening wavelengths and then change of angle loses the influence.

Figure 3.15: Sensitivity due to the dielectric (a) (b) contrast change (hh) Figure 3.16: Ground b.s. in the presence of snow; (a) vv polarization, (b) hh polarization Therefore, according to figure 3.16 we can finally conclude that the backscattering sensitivity to the snow density in the C-band comes from the sensitivity to the change of ground dielectric properties. Therefore, presented semi-empirical model, derived by suppressing sensitivity to the ground dielectric properties, can hardly be applied in C-band without being modified significantly. On the other side, a model primarily sensitive to any change of ground dielectric constant can eventually be useful.

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Information collected in C-band can not be used independently for depth estimation, because sensitivity is quite small, but as it will be shown through model exposed in III.3 by combining it will information from other frequency bands, we can acquire some information about snow cover depth. III.2.3 X – band The sensitivity studies in X-band (8 – 12.5 GHz) are done for 10 GHz frequency value.

Figure 3.17: Rayleigh model; (left) vv polarization, (right) hh polarization The differences which can be noticed according to previous bands are: -

Significantly increased density sensitivity and quite increased depth sensitivity. Obviously different angular dependence for two polarization channels. Slightly bigger depth sensitivity in case of using Rayleigh model.

Figure 3.18: QCA model; (left) vv polarization, (right) hh polarization By analyzing figure 3.19, we can notice the same kind of change relative to the C-band, like it was for that band relative to the L-band. Increase of frequency is causing increase of air-snow backscattering component and decrease of volume-ground interaction component. The consequence of the frequency increase, extinction increase, causes enhance of the volume backscattering component. It is also causing increase of volume-ground interaction component, but the previously 36


Nikola BEŠIĆ

mentioned decrease effect is the more dominant one. As it can be seen, even in this band the most dominant component is the ground backscattering component. Increased density sensitivity, as it will be shown, can again be explained by analyzing ground backscattering and the way change of the density affects it.

(a) (b) Figure 3.19: QCA model; (a) vv polarization, (b) hh polarization Using 3.19 the second observation can be also justified. By observing angular dependence of airsnow component for two polarizations, we can notice significant drop for hh polarization around 55 degrees, that doesn’t exist in the case of vv polarization, which is the consequence of the Brewster angle appearing in the vv case and causing this sort of angular dependence. This effect doesn’t exist in the same form for the ground backscattering component, because of the shorter wavelength and refracted angle effects which are changing the surface backscattering shape. By recalling the fact that Fresnel power transmission as well as angular propagation effect are not functions of frequency, the increased sensitivity in this direction must come either from extinction coefficient effect or from the ground backscattering in the presence of snow. Extinction coefficient can be excluded according to figures 3.17 and 3.18 because we have sensitivity increase which is directed the same for both of the models. For QCA model in case of significant extinction influence sensitivity would be decreased, not increased. Therefore, like in the case of previous two bands sensitivity is determined by the influence of density to the ground backscattering (figure 3.20).

(a) (b) Figure 3.20: (hh) Influence of separate terms on ground backscattering; (a) Rayleigh, (b) QCA

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If we recall the conclusion based on figure 3.15, that for this range of the dielectric contrast changes, the affect of the change to the ground backscattering is independent of frequency, the only possible conclusion could be that effect of wavelength shortening and angle change has change its nature. In X-band, this effect causes decrease of backscattering with density increase, like it is shown in the following figure:

(a) (b) Figure 3.21: Ground backscattering in the presence of snow;(a) vv polarization, (b) hh polarization This effect is the consequence of surface backscattering frequency dependence. Like it is already mentioned and can be seen in Appendix I.4 with increasing frequency we are decreasing backscattering angular dependence and approaching to the almost isotropic function. However, due to the fact that for smaller frequencies we don’t have a linear decrease of backscattering with angle increase, this backscattering function shows opposite behavior, its slope changes the sign and we have slight increase with the increase of angle. The whole effect is being increased by shortening wavelength due to the medium, because it is equal as increasing incident frequency. Then, bigger refracted angle means smaller value, and thus bigger density leads to the smaller backscattering. Finally, we can conclude that in X-band ground backscattering as the dominant component and therefore the whole backscattering are sensitive to the snow density change due to the change of ground dielectric properties visible to the incident waves and due to the change in angle and wavelength shortening. This conclusion leads to the possibility that existing semi-empirical model for L-band, sensitive only to the angle and incident wavelength maybe can be used with certain modification at X-band frequencies.

(a) (b) Figure 3.22: vv polarization, Depth sensitivity: (a) Rayleigh model, (b) QCA model 38


Nikola BEŠIĆ

Increased depth sensitivity is consequence of increased volume backscattering component and partly consequence of increased depth sensitivity of the ground backscattering component. Both of the effects are due to the extinction increase in X-band. Rayleigh model shows bigger sensitivity because of the bigger extinction for corresponding density relative to the QCA model. Concerning density estimation, this fact can be good, because QCA model is definitely more reliable one at 10 GHz frequency and low depth sensitivity can make estimation of density using only X-band data. On the other side, for depth estimation it is impossible to use only this data for depth estimation. It is necessary to combine it with the data from other bands.

III.3 Statistical approach In order to confirm and better illustrate previous conclusion, we have conducted several statistical analysis. They are therefore also treating backscattering sensitivity to snow density and depth, but also sensitivity to one possibly critical but still necessary assumption, radius of the spherical particle. At the end, appropriate backscattering coefficient histograms are present. Those can be used for practical validation of this study as soon as we can get snow measurements and appropriate radar acquisition results. First, we have assumed that dry snow density is following normal distribution centered around 200 kg/m3 with standard deviation of 16 kg/m3. For each density sampled we were varying depth according to uniform distribution (values between 30 cm and 170 cm). Appropriate backscattering coefficient mean values for one density value and various depth values as well as its standard deviation are shown in the following figures. The analysis was done for the same ground and snow surface parameters, same particle radius and ground dielectric properties assumed in the III.2. For angle value, acquisition standard of 40 degrees is taken.

Figure 3.23: QCA model, hh polarization; depth sensitivity statistical analysis It can be seen that increase in density causes increase in backscattering in L-band, like it is already stated and justified, while decrease of the same in the C and especially in the X-band. Depth sensitivity is increasing with frequency, but even in X-band it is almost insignificant. Difference in slope of backscattering coefficient standard deviation as the function of density for two models (figures 3.23 and 3.24) can be justified by the fact that the depth sensitivity comes from volume backscattering component, for all three frequency bands. With ignored multiple scattering, bigger density means bigger volume backscattering and therefore bigger backscattering standard 39


Nikola BEŠIĆ

deviation due to the depth variation. With included multiple scattering effect, dependence is opposite.

Figure 3.24: Rayleigh model, hh polarization; depth sensitivity statistical analyzis

Figure 3.25 shows the way radius change affects backscattering coefficient. Radius variation is achieved by uniform distribution of sphericity and dendricity (eq. 3.22) along the whole possible range (between 0 and 1). Certainly, with frequency increase, sensitivity on particle size is increasing, because we have more significant volume interaction. On the other side, it can be seen that density change doesn’t affect this sensitivity too much, even though we have taken multiple scattering into the account (QCA model). Conclusion could be that the sensitivity is quite small, and that our assumption of ideal spherical particles can be taken as at least satisfying one.

Figure 3.25: QCA, hh polarization; particle radius sensitivity statistical analyzis Backscattering coefficient histograms for varying density, depth, radius (according to already mentioned distributions) and for varying angle (uniform distribution between 38 and 42 degrees) are given in the figures 3.26 and 3.27.

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Obvious fact that the backscattering for this range of angles and snow cover parameters is the biggest in C-band can be justified by ground backscattering angular distribution. This and the fact that hh polarization gives significantly lower backscattering in each of the bands, can be used as the future acquisitions guideline.

Figure 3.26 : Backscattering coefficient histograms (vv polarization) for L, C and X band

Figure 3.27 : Backscattering coefficient histograms (hh polarization) for L, C and X band

III.4 Semi – empirical model Semi-empirical model for dry snow, derived by Shi and Dozier [1,2] will be briefly described in this part. As the SWE is the function of two independent variables: density and depth, the separated models for both of the parameters will be presented.

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Nikola BEŠIĆ

III.4.1 Dry snow density estimation using L-band data Generally, relation backscattering coefficients of a surface in vv and hh co-polarized channels is a function of angle, frequency and dielectric and roughness properties of the ground. With dielectric constant decrease and ground roughness increase difference is decreasing, while with increase of angle and frequency we difference decrease. Led by the same conclusion we have reached in the previous part, that backscattering in both of the channels is more sensitive to the angle and incident wavelength change than to the ground dielectric properties in the L-band, Shi and Dozier have developed model which is increasing sensitivity of difference to the angle and frequency, while decreasing the same according to dielectric and roughness ground properties. 𝑔𝑔

𝑔𝑔

After the simulation of 𝜎𝜎ℎℎ and 𝜎𝜎𝑣𝑣𝑣𝑣 using IEM (Integral Equation model – Appendix I.4) over a wide range of incidence angles (15o – 70o) and wave numbers (0.285 cm-1 to 0.385 cm-1), dielectric and roughness conditions (moisture (2%-50%), RMS height (2mm – 50mm) and correlation length (25mm-350mm)) and snow densities (100 kg m-3 – 550 kg m-3), the following relationship (8) has been established. 𝑔𝑔

𝑔𝑔

log10 ��𝜎𝜎ℎℎ + �𝜎𝜎𝑣𝑣𝑣𝑣 �

𝑔𝑔

𝑔𝑔

𝑔𝑔

= 𝑎𝑎(𝜃𝜃𝑟𝑟 , 𝑘𝑘𝑠𝑠 ) + 𝑏𝑏(𝜃𝜃𝑟𝑟 , 𝑘𝑘𝑠𝑠 ) log10 �𝜎𝜎ℎℎ + 𝜎𝜎𝑣𝑣𝑣𝑣 � + 𝑐𝑐 (𝜃𝜃𝑟𝑟 , 𝑘𝑘𝑠𝑠 ) log10 𝜎𝜎ℎℎ +

𝑔𝑔 𝜎𝜎 𝑑𝑑(𝜃𝜃𝑟𝑟 , 𝑘𝑘𝑠𝑠 ) log10 � ℎℎ 𝑔𝑔 � 𝜎𝜎𝑣𝑣𝑣𝑣

+

𝑔𝑔 2 𝜎𝜎ℎℎ 𝑒𝑒(𝜃𝜃𝑟𝑟 , 𝑘𝑘𝑠𝑠 ) log10 � 𝑔𝑔 � 𝜎𝜎𝑣𝑣𝑣𝑣

(3.25)

Given parameters, which are only angle 𝜃𝜃𝑟𝑟 and wavenumber 𝑘𝑘𝑠𝑠 functions, are deduced from the equation: {𝑎𝑎, 𝑏𝑏, 𝑐𝑐, −𝑑𝑑, 𝑒𝑒}(𝜃𝜃𝑟𝑟 , 𝑘𝑘𝑠𝑠 )

3

= 𝑒𝑒𝑒𝑒𝑒𝑒 �𝑆𝑆1 sin 𝜃𝜃𝑟𝑟 + � 𝐴𝐴𝑛𝑛 cos n 𝜃𝜃𝑟𝑟 � 𝑛𝑛=0

4

(3.26)

× �1 + (𝑘𝑘𝑑𝑑 − 1)𝑒𝑒𝑒𝑒𝑒𝑒 �𝐿𝐿1 sin 𝜃𝜃𝑟𝑟 log 𝑘𝑘𝑑𝑑 + � 𝐵𝐵𝑛𝑛 sinn 𝜃𝜃𝑟𝑟 ��

where 𝑘𝑘𝑑𝑑 = 𝑘𝑘

𝑘𝑘 𝑠𝑠

0 �𝜀𝜀 𝑚𝑚𝑚𝑚𝑚𝑚

𝑛𝑛=0

with 𝜀𝜀𝑚𝑚𝑚𝑚𝑚𝑚 as the dielectric constant corresponding to the minimal dry snow

density assumed of 100 kg/m3, while all the other coefficients are simulation results and are given through the appropriate table in the Appendix II.1.

Exponential term is giving coefficients corresponding to the minimal dry snow density wavenumber, while the other term represents correction factor for the change of snow density and thus incident wavenumber. This model shows very good accuracy (figure 3.28), especially for the bigger angles, which is coexistent with our theoretical analysis, where it is possible to see that particularly hh channel and by that difference between channels, is more sensitive (positively) to the density change at the bigger angles. By adding absolute and relative SAR calibration error, it is also concluded that absolute error doesn’t affect model accuracy, while concerning relative one, better accuracy is achieved for the bigger angles. Therefore, this model should be used for angles above 40 degrees. 42


Nikola BEŠIĆ

Figure 3.28: (solid curve) Error Root Mean Square relative to IEM model, (dotted curve) relative calibration error included. Capture from [1], page 2468, fig. 2 By properly assuming that extinction is negligible and that therefore ground backscattering component is not attenuated by the layer. Also, volume, volume-ground interaction components can be neglected due to the same reason, while low frequency and low dielectric constant make air snow also not significantly influential. Therefore, it is possible to express total backscattering using ground backscattering in the presence of snow and Fresnel power transmission coefficients: 𝑔𝑔

𝑡𝑡 2 𝜎𝜎𝑝𝑝𝑝𝑝 (𝑘𝑘0 , 𝜃𝜃𝑖𝑖 ) = 𝑇𝑇𝑝𝑝𝑝𝑝 (𝜃𝜃𝑖𝑖 )𝜎𝜎𝑝𝑝𝑝𝑝 (𝑘𝑘𝑠𝑠 , 𝜃𝜃𝑟𝑟 )

(3.27)

By integrating model represented by 3.25 into the previous equation, we are getting final model for dry snow density estimation. Solving equation 3.28 gives us snow dielectric constant, which can be then used for density calculation using one of the derived empirical formulas (for example – 2.3).

log10 �

𝑡𝑡 𝑡𝑡 �𝜎𝜎ℎℎ �𝜎𝜎𝑣𝑣𝑣𝑣 + � 𝑇𝑇ℎℎ (𝜃𝜃𝑖𝑖 , 𝜀𝜀𝑠𝑠 ) 𝑇𝑇𝑣𝑣𝑣𝑣 (𝜃𝜃𝑖𝑖 , 𝜀𝜀𝑠𝑠 ) = 𝑎𝑎(𝜃𝜃𝑖𝑖 , 𝑘𝑘0 , 𝜀𝜀𝑠𝑠 )

𝑡𝑡 𝑡𝑡 𝜎𝜎ℎℎ 𝜎𝜎𝑣𝑣𝑣𝑣 + 𝑏𝑏(𝜃𝜃𝑖𝑖 , 𝑘𝑘0 , 𝜀𝜀𝑠𝑠 ) log10 � 2 + 2 � 𝑇𝑇ℎℎ (𝜃𝜃𝑖𝑖 , 𝜀𝜀𝑠𝑠 ) 𝑇𝑇𝑣𝑣𝑣𝑣 (𝜃𝜃𝑖𝑖 , 𝜀𝜀𝑠𝑠 ) 𝑡𝑡 𝑡𝑡 2 (𝜃𝜃 𝑇𝑇𝑣𝑣𝑣𝑣 𝜎𝜎ℎℎ 𝜎𝜎ℎℎ 𝑖𝑖 , 𝜀𝜀𝑠𝑠 ) � + 𝑑𝑑(𝜃𝜃𝑖𝑖 , 𝑘𝑘0 , 𝜀𝜀𝑠𝑠 ) log10 � 𝑡𝑡 2 � + 𝑐𝑐(𝜃𝜃𝑖𝑖 , 𝑘𝑘0 , 𝜀𝜀𝑠𝑠 ) log10 � 2 𝜎𝜎𝑣𝑣𝑣𝑣 𝑇𝑇ℎℎ (𝜃𝜃𝑖𝑖 , 𝜀𝜀𝑠𝑠 ) 𝑇𝑇ℎℎ (𝜃𝜃𝑖𝑖 , 𝜀𝜀𝑠𝑠 ) 2 𝑡𝑡 2 (𝜃𝜃 𝜎𝜎ℎℎ 𝑇𝑇𝑣𝑣𝑣𝑣 𝑖𝑖 , 𝜀𝜀𝑠𝑠 ) + 𝑒𝑒(𝜃𝜃𝑖𝑖 , 𝑘𝑘0 , 𝜀𝜀𝑠𝑠 ) log10 � 𝑡𝑡 2 � 𝜎𝜎𝑣𝑣𝑣𝑣 𝑇𝑇ℎℎ (𝜃𝜃𝑖𝑖 , 𝜀𝜀𝑠𝑠 )

(3.28)

Validation procedure for this model has showed overall RMSE of 11.3% and maximum relative error of 24.3 %. The same reference [1] gives methods of ground dielectric constant estimation and roughness properties estimation in L-band. III.4.2 Dry snow depth estimation using C and X band data As it is shown, concerning depth sensitivity, we can not simply, like in the case of density sensitivity, neglect all the other components and observe just the dominant one. First, because of the fact that core of the sensitivity lies in the less significant components and the second, because of the fact that we have increased sensitivity in the C and X band where those components are playing more important role and can not be simply neglected.

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Generally, total backscattering is, like it can be seen in II.2 part, function of 13 input parameters. Therefore, it is almost impossible to calculate exactly the ones we need to SWE because we must have a lot of backscattering data and also a lot of other input parameters well known. It is necessary to build models for separate components and make them sensitive to less parameters. This model, assuming that snow density and ground dielectric and roughness properties are known from L-band measurements, is reducing number of unknowns from 13 to 3, which can be calculated using C and X band polarimetric data. Each subcomponent is treated separately. 1. Air-snow subcomponent Like it is obvious from III.2 part snow depth is not affecting this component at all. Therefore, it doesn’t contain any useful information concerning snow cover depth. Figure 3.29 shows error when we totally neglect this component and the same one if we calculate it using IEM with a priori knowledge of likely roughness properties and with snow dielectric constant derived from L-band.

𝑎𝑎 Fig. 3.29: (A) RE - ignoring of 𝜎𝜎𝑝𝑝𝑝𝑝 , (B) RE – IEM model (16). Capture from [2], Fig.2, page 2481.

Conclusion is, that if we are using IEM estimation with assumed input data about the snow cover RMS height, correlation length and with derived snow dielectric properties, we can for bigger angles quite successfully eliminate this component. 2. Vol ume – ground subcomponent During our analysis, as it is introduced, we didn’t consider the last two cases mentioned in II.2 (non-coherent components). Unlike the backscattering which is consequence of redirected specular reflection (coherent component), and which decreases with frequency increase, those two cases are getting significant at higher frequencies (figure 3.30), and which is especially important, they contain information concerning depth. Therefore, model for estimating this component sensitive only to known data (snow density, frequency, angle, roughness properties) and optical depth and albedo is derived. For vertical vv co-polarization: 𝑔𝑔𝑔𝑔

𝑏𝑏(𝑓𝑓,𝜃𝜃𝑟𝑟 )

2 (𝑓𝑓, 𝜎𝜎�𝑣𝑣𝑣𝑣 = 𝑎𝑎(𝑓𝑓, 𝜃𝜃𝑟𝑟 )𝑇𝑇𝑣𝑣𝑣𝑣 𝜃𝜃𝑟𝑟 )𝑅𝑅𝑣𝑣𝑣𝑣

(𝑓𝑓, 𝜃𝜃𝑟𝑟 )𝑘𝑘𝑠𝑠 (𝑓𝑓)𝑑𝑑 ∙ exp[𝑐𝑐(𝑓𝑓, 𝜃𝜃𝑟𝑟 )𝑘𝑘1 𝑠𝑠 + 𝑑𝑑(𝑓𝑓, 𝜃𝜃𝑟𝑟 )𝑘𝑘𝑒𝑒 (𝑓𝑓)𝑑𝑑] 3

{𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑 }(𝜃𝜃𝑟𝑟 ) = � 𝐴𝐴𝑛𝑛 𝜃𝜃𝑟𝑟𝑛𝑛 𝑛𝑛=0

44

(3.29a) (3.29b)


Nikola BEŠIĆ

𝑔𝑔𝑔𝑔

Fig. 3.30: (A) Relative contribution of 𝜎𝜎𝑝𝑝𝑝𝑝 , (B) Relative error in case of ignoring. Capture from [2], Fig.1, page 2479. For horizontal co-polarization, separate formulas for coherent and non-coherent component are derived: 𝑔𝑔𝑔𝑔𝑔𝑔

2 𝜎𝜎�ℎℎ = 𝑇𝑇ℎℎ 𝜔𝜔𝑅𝑅ℎℎ exp(−(2𝜇𝜇𝑟𝑟 𝑘𝑘1 𝑠𝑠)2 ) exp �−

In case k 1 s3, 𝑔𝑔𝑔𝑔𝑔𝑔

𝜎𝜎�ℎℎ

{𝑎𝑎, log(𝑏𝑏) , 𝑐𝑐, log(𝑑𝑑) , 𝑒𝑒} = � 𝐴𝐴𝑛𝑛 𝜃𝜃𝑟𝑟𝑛𝑛 𝑛𝑛=0

𝑏𝑏(𝜃𝜃𝑟𝑟 )

2 (𝜃𝜃 )𝑎𝑎(𝜃𝜃 )𝑅𝑅 = 𝑇𝑇ℎℎ 𝑖𝑖 𝑟𝑟 ℎℎ

(𝜃𝜃𝑟𝑟 )𝜔𝜔𝑘𝑘𝑒𝑒 𝑑𝑑 ∙ exp[�𝑐𝑐(𝜃𝜃𝑟𝑟 ) + 𝑑𝑑(𝜃𝜃𝑟𝑟 )𝑅𝑅ℎℎ (𝜃𝜃𝑟𝑟 )�𝑘𝑘𝑒𝑒 𝑑𝑑] 3

{𝑎𝑎, log(𝑏𝑏) , 𝑐𝑐, d, 𝑒𝑒} = � 𝐴𝐴𝑛𝑛 𝜃𝜃𝑟𝑟𝑛𝑛 𝑛𝑛=0

(3.30)

(3.31a)

(3.31b)

(3.32a)

(3.32b)

The appropriate non coherent model is chosen according to the surface roughness (k 1 s value, which represents the ratio between RMS height of the surface and the incidence wavelength). All the coefficient are given in the Appendix II.2. Improvement is shown through the following figure:

𝑔𝑔𝑔𝑔

Fig. 3.31: Relative error of 𝜎𝜎�𝑝𝑝𝑝𝑝 . Capture from [2], fig.1, page 2479. 45


Nikola BEŠIĆ

3. Ground backscattering subcomponent Concerning this component, the goal is to deduce the relations between ground backscattering 𝑔𝑔 𝑔𝑔 𝑔𝑔 coefficients 𝜎𝜎𝑉𝑉𝑉𝑉 and 𝜎𝜎𝐻𝐻𝐻𝐻 in C-band and 𝜎𝜎𝑉𝑉𝑉𝑉 in X-band, by using data which are collected during the L-band measurements of the same area– ground dielectric constant 𝜀𝜀𝑔𝑔 and Root Mean Square (RMS) height of a surface s. 𝑔𝑔 𝑔𝑔 Relation between two channels at the same frequency 𝜎𝜎𝑉𝑉𝑉𝑉 /𝜎𝜎𝑉𝑉𝑉𝑉 is mostly dependent on s and 𝜀𝜀𝑔𝑔 .. It increases with increasing s, while decreases with increasing 𝜀𝜀𝑔𝑔 . By using IEM simulated data over wide range of surface properties, model which gives the ratio as the function of incidence angle and given L-band data is developed (3.33). 𝑘𝑘1 𝑠𝑠 ⎧exp �𝑎𝑎(𝜃𝜃𝑟𝑟 ) + 𝑏𝑏(𝜃𝜃𝑟𝑟 )𝑘𝑘1 𝑠𝑠 + 𝑐𝑐(𝜃𝜃𝑟𝑟 )(𝑘𝑘1 𝑠𝑠)2 + 𝑑𝑑(𝜃𝜃𝑟𝑟 ) log � � log(Γ0 )� , k1 s < Ct 𝑔𝑔 𝜎𝜎𝑉𝑉𝑉𝑉 (𝜃𝜃𝑟𝑟 ) 𝐶𝐶𝑡𝑡 = 𝑔𝑔 2.62 𝜎𝜎𝐻𝐻𝐻𝐻 (𝜃𝜃𝑟𝑟 ) ⎨ 1, k1 s ≥ Ct , Ct = ⎩ cos 0.825 𝜃𝜃𝑟𝑟 2

4

𝑛𝑛=1

𝑛𝑛=0

{𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑, 𝑒𝑒}(𝜃𝜃𝑟𝑟 ) = � 𝐿𝐿𝑛𝑛 sinn 𝜃𝜃𝑟𝑟 + � 𝐵𝐵𝑛𝑛 cosn 𝜃𝜃𝑟𝑟

(3.33a)

(3.33b)

For relations between coefficients at different bands, the situation is a bit more complicated because the RMS height is not the only surface factor which has a big influence on ratio, but also the surface correlation length. The derived empirical model is given in the following relation (3.34). Coefficients are given in the Appendix II.3. 𝑔𝑔

𝑔𝑔

log[ 𝜎𝜎𝑉𝑉𝑉𝑉 (𝑋𝑋, 𝜃𝜃𝑟𝑟 )] = 𝑎𝑎(𝜃𝜃𝑟𝑟 ) + 𝑏𝑏(𝜃𝜃𝑟𝑟 )log[𝜎𝜎𝑉𝑉𝑉𝑉 (𝐶𝐶, 𝜃𝜃𝑟𝑟 )] + 𝑐𝑐(𝜃𝜃𝑟𝑟 )log[𝑘𝑘1 𝑠𝑠(𝐶𝐶)] + 𝑑𝑑(𝜃𝜃𝑟𝑟 )log 2 [𝑘𝑘1 𝑠𝑠(𝑋𝑋) − 𝑘𝑘1 𝑠𝑠(𝐶𝐶)] , 𝑓𝑓𝑓𝑓𝑓𝑓 𝑘𝑘1 𝑠𝑠 < 2.3 𝑔𝑔

𝑔𝑔

𝑔𝑔

log[ 𝜎𝜎𝑉𝑉𝑉𝑉 (𝐶𝐶, 𝜃𝜃𝑟𝑟 )] = 𝑎𝑎(𝜃𝜃𝑟𝑟 ) + 𝑏𝑏(𝜃𝜃𝑟𝑟 )log[𝜎𝜎𝑉𝑉𝑉𝑉 (𝑋𝑋, 𝜃𝜃𝑟𝑟 )] + 𝑐𝑐(𝜃𝜃𝑟𝑟 )log[𝜎𝜎𝑉𝑉𝑉𝑉 (𝐿𝐿, 𝜃𝜃𝑟𝑟 )] + 𝑑𝑑(𝜃𝜃𝑟𝑟 )log 2 [𝑘𝑘1 𝑠𝑠(𝑋𝑋) − 𝑘𝑘1 𝑠𝑠(𝐶𝐶)] + 𝑒𝑒(𝜃𝜃𝑟𝑟 ) log Γ0 , 𝑓𝑓𝑓𝑓𝑓𝑓 𝑘𝑘1 𝑠𝑠 ≥ 2.3

(3.34a)

(3.34b)

Fig. 3.32: Root Mean Square Errors of given models. Capture from [2], fig. 3, page 2482, fig.4, page 2483.

The figure above shows that this model accuracy is satisfying, especially in case of X-band.

4. Volume backscattering component The last step before proceeding to snow pack depth estimation is to establish the relation between 𝑘𝑘 albedos (ratio between scattering and extinction coefficient: 𝜔𝜔 = 𝑠𝑠 ) at the two available 𝑘𝑘 𝑒𝑒

frequencies – C and X bands (3.35). This is done by using the simulation of volume scattering component (13) over wide range of input parameters, as well as for the ground and volume-ground interaction backscattering components. 46


Nikola BEŠIĆ

log 𝜔𝜔(𝑋𝑋) = 1.334 + 0.9929 log 𝜔𝜔(𝐶𝐶) − 1.9623𝜔𝜔(𝐶𝐶) + 0.7587𝜔𝜔2 (𝐶𝐶)

(3.35)

After developing all those models, the number of unknown variables from equations in II.2 is reduced to three: one of the ground backscattering coefficients (C or X, HH or VV), albedo 𝜔𝜔 and optical thickness 𝜏𝜏 (𝜏𝜏 = 𝑘𝑘𝑒𝑒 𝑑𝑑). Those three variables can be solved numerically using the following 𝑡𝑡 (𝐶𝐶), 𝑡𝑡 (𝐶𝐶), 𝑡𝑡 SAR data: 𝜎𝜎𝑣𝑣𝑣𝑣 𝜎𝜎ℎℎ 𝜎𝜎𝑣𝑣𝑣𝑣 (𝑋𝑋).

In order to estimate the depth from optical thickness, it is necessary to express the extinction coefficient as function of the derived parameters. It is done by using the equations for dry snow absorption coefficients ratio (3.36). 𝑘𝑘𝑎𝑎 (𝑋𝑋) 𝜏𝜏(𝑋𝑋)[1 − 𝜔𝜔(𝑋𝑋)] = 𝑘𝑘𝑎𝑎 (𝐶𝐶) 𝜏𝜏(𝑋𝑋)[1 − 𝜔𝜔(𝑋𝑋)] 𝜌𝜌 𝑘𝑘𝑎𝑎 (𝑋𝑋) � − 3.4217log( 𝑘𝑘𝑎𝑎 (𝑋𝑋) = 1.334 + 1.2182 log � ) 0.917 𝑘𝑘𝑎𝑎 (𝐶𝐶)

(3.36a) (3.36b)

Finally, the depth is expressed as:

𝑑𝑑 =

𝜏𝜏(𝑋𝑋)[1 − 𝜔𝜔(𝑋𝑋)] 𝑘𝑘𝑎𝑎 (𝑋𝑋)

47

(3.37)


Nikola BEŠIĆ

IV Wet snow analysis Wet snow is snow above 0 oC, when we can not assume existence of only ice particles distributed in the air, but also existence of liquid water, which is the consequence of the ice melting process. Theory used to describe wet snow layer influence on EM wave propagating through it is briefly described in the beginning of this chapter. Next, using this theory we have done certain simulations for different microwave bands and by explaining them we have pointed to the main difficulties in wet snow monitoring using SAR technique. Also, multilayer snow cover model and its problematics is presented.

IV.1 QCA – CP model Quasi Crystalline Approximation with Coherent Potentials is modification of the QCA theory which was introduced in order to analyze dry snow cover. The main difference is that according to QCA with coherent potentials, as multiple scattering becomes more important, coherent wave “sees” not host medium like in the QCA case, but new average medium. This fact allows its application on mediums with a big difference between host and particle dielectric constant, or simply with a big variation of dielectric properties. If we recall water dielectric properties from II.1, we can see that it makes it perfect for wet snow layer application. Real part of the effective medium dielectric constant, normalized to vacuum dielectric constant, in case of the spherical particles is given by the relation: 𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 ′ = 1 +

3𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 (𝜀𝜀𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 − 1)𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 0

3𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 0 + �1 − 𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 ��𝜀𝜀𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 − 1�

(4.1)

Where 𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 0 represents effective dielectric constant of a medium when multiple scattering effects are neglected, and is defined by relation: 𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 0 = 1 +

3𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 (𝜀𝜀𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 − 1)𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 0

3𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 0 + �1 − 𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 ��𝜀𝜀𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 − 1�

(4.2)

Effective dielectric constant imaginary part, which introduces multiple coherent scattering effect is given by: 𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 ′′

3

9𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 (𝜀𝜀𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 − 1)2 𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 0 �𝑘𝑘√𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 0 � 𝜐𝜐0 = 2 𝑆𝑆0 2𝜋𝜋 �3𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 0 + �1 − 𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 �(𝜀𝜀𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 − 1)�

(4.3)

Like in the case of QCA model, wavenumber can be expressed as: 𝐾𝐾 = 𝐾𝐾 ′ + 𝑗𝑗𝑗𝑗 ′′ =

and extinction coefficient is:

2𝜋𝜋 2𝜋𝜋 𝑅𝑅𝑅𝑅 ��𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 ′ + 𝑗𝑗𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 ′′ � + 𝑗𝑗 𝐼𝐼𝐼𝐼 ��𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 ′ + 𝑗𝑗𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 ′′ � 𝜆𝜆 𝜆𝜆

49

(4.4)


Nikola BEŠIĆ

𝜅𝜅𝑒𝑒 = 2𝐾𝐾 ′′ =

4𝜋𝜋 𝐼𝐼𝐼𝐼 ��𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 ′ + 𝑗𝑗𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 ′′ � 𝜆𝜆

(4.5)

Unlike, the previous assumption, where we were claiming that absorption can be neglected, this time, because of particles imaginary part which is, as it will be shown, not negligible, we must introduce relation for scattering coefficient. By doing that, we can simply calculate absorption coefficient and albedo. Then, we can claim that we have all the necessary radiative transfer equation parameters. 2

𝑘𝑘 2 (𝜀𝜀𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 − 1)𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 0 3𝑣𝑣0 𝜅𝜅𝑠𝑠 = 𝑓𝑓 � � 𝑆𝑆 2𝜋𝜋 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 3𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 0 + �1 − 𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 �(𝜀𝜀𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 − 1) 0

(4.6)

Liquid water content is introduced through equivalent particles. Ice particles are replaced by particles which are ice spheres surrounded by thin water film. Their radius, which defines volume, is given by: 𝑟𝑟𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑟𝑟𝑖𝑖𝑖𝑖𝑖𝑖

where 𝑓𝑓𝑙𝑙𝑙𝑙 represents liquid water volume fraction.

1

𝑓𝑓𝑙𝑙𝑙𝑙 3 �1 + � 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖

(4.7)

Equivalent dielectric constant is given by Rayleigh mixing formula: 𝜀𝜀𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 − 1 (𝜀𝜀𝑙𝑙𝑙𝑙 − 1)(𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 + 2𝜀𝜀𝑙𝑙𝑙𝑙 ) + 𝛾𝛾(𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 − 𝜀𝜀𝑙𝑙𝑙𝑙 )(1 + 2𝜀𝜀𝑙𝑙𝑙𝑙 ) = 𝜀𝜀𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 + 1 (𝜀𝜀𝑙𝑙𝑙𝑙 + 2)(𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 + 2𝜀𝜀𝑙𝑙𝑙𝑙 ) + 2𝛾𝛾(𝜀𝜀𝑖𝑖𝑖𝑖𝑖𝑖 − 𝜀𝜀𝑙𝑙𝑙𝑙 )(𝜀𝜀𝑙𝑙𝑙𝑙 − 1) 3 𝑓𝑓𝑙𝑙𝑙𝑙 −1 𝑟𝑟𝑖𝑖𝑖𝑖𝑖𝑖 𝛾𝛾 = � � = �1 + � 𝑟𝑟𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖

(4.8a) (4.8b)

So, now it is possible to present wet snow cover radiative transfer behavior as a function of liquid water content, with assumed constant ice volume fraction.

Figure 4.1: Radiative transfer equation parameters of a wet snow cover Unlike the QCA case, now we don’t have extinction decrease for general density increase, caused by liquid water volume fraction increase. But, it is important to notice that albedo is increasing, which signifies that scattering is still rising more rapidly than absorption. 50


Nikola BEŠIĆ

IV.2 Simulation analysis During our studies we simulated wet snow cover electromagnetic behaviour using previously presented QCA-CP model by introducing water through the modified mixing particles according to the given Rayleigh mixing formula. Because of bigger density and big equivalent particles dielectric contrast it doesn't make any sense to use Rayleigh model in parallel like for a dry snow. Ground is again, like in the case of a dry snow, defined using isotropic exponential correlation function with correlation length of 5 cm and with RMS of 1 cm. Due to the fact that air-snow component now, as it will be seen, plays significant role in the backscattering, air-snow interface is described with the same correlation function and properties, but with a RMS of 0.5 cm, because it is reasonable to assume that air surface is a bit more smooth than the corresponding ground one, which is also the case with a dry snow, but back there is is not so important Ground dielectric constant is 6. Ice particles are described with ice dielectric constant value 3.15, radius – 0.225 mm and with volume fraction of 0.2, which defines appropriate dry snow density, expanded by liquid water presence. Sensitivity analysis are done in three bands (L, C and X) for various snow densities, or basically because ice volume fraction is considered as constant, for various liquid water contents and for various snow cover depths. IV.2.1 L-band The sensitivity studies in L-band (1 – 2 GHz) are done for 1.5 GHz frequency value.

Figure 4.2: QCA-CP model; vv polarization The primary observations concerning figures 4.2 and 4.3 are: - Quite significant density sensitivity, while on the other side very low depth sensitivity. - Somehow unusual density sensitivity, first there is decrease with increasing density, or liquid water content, and then there is a quite big increase. - If we recall appropriate figures from Chapter III (figures 3.4 and 3.5) we can notice that backscattering for approximately same snow and ground properties (difference is in air surface RMS height, but it can be neglected because in case of a dry snow air surface RSM doesn't affect total backscattering too much) in case of a liquid water present, backscattering is significantly lower (-10 dB). 51


-

Nikola BEŠIĆ

There is certain difference in density sensitivity for vv and hh polarization. For bigger angles, in case of hh polarization we have the biggest backscattering for smaller liquid water contents, which is not the case with a vv polarization.

Figure 4.3: QCA-CP model; hh polarization In order to justify those observations and make some conclusions, it is necessary to examine separately behavior of the backscattering components.

(a)

(b)

Figure 4.4: QCA-CP model, density sensitivity; (a) vv polarization; (b) hh polarization Concerning volume and volume-ground interaction components, like in the case of a dry snow, they can be considered as negligible. Here it is due to the fact that they are proportional to snow layer albedo, and this one is quite small in case of a liquid water presence, because of the big absorption effect. Volume component is increasing with density increase, which can be explained by the fact that extinction and albedo are increasing, we have bigger absorption but also bigger volume scattering, where the later one increases more rapidly (figure 4.1). On the other side, volume-ground is decreasing with density increase, which can be justified by the fact that waves which are travelling all the way to the ground and then after interaction with snow all the way back are being attenuated more with bigger liquid water content. Ground backscattering and air-snow backscattering components are the dominant one. Their big density sensitivity is causing the big sensitivity of the total backscattering component, while the fact that their sensitivity directions are opposite can explain mentioned unusual sensitivity behavior. For smaller densities ground backscattering is bigger than air snow one, so we have decrease of total backscattering with density increase until the air snow component becomes bigger when total 52


Nikola BEŠIĆ

backscattering starts increasing with density. This explanation is confirmed by the fourth observation, ground backscattering, which is dominant for smaller densities has a smaller angle slope than the corresponding air-ground because of the difference in RMS height and especially because of the angle refraction in case of a ground component which reduces slope in case of hh polarization (figure 3.8). Sensitivity direction for air-snow component is consequence of snow dielectric constant, which is increasing with density increase, but it is interesting to estimate which term is dominantly causing this kind of ground backscattering sensitivity. If we recall figure 3.9 we can immediately conclude that Fresnel power transmission, extinction and dielectric contrast are having bigger impact than wavelength shortening and angle refraction. By doing analysis which resulted with figure 4.5 we have determined: - Fresnel transmission contributes to this direction, because with increasing density we have bigger dielectric contrast at air-snow interface. - Dielectric contrast has a bigger influence than wavelength shortening and angle refraction, which can be explained by the angular isotropy caused by quite big wavelength shortening. This was not a case with a dry snow in L-band, because the wavelength shortening was not so big to cause significant change of angular slope. - This time the biggest influence on sensitivity comes from a extinction coefficient change. Finally, we can claim that most of the ground component sensitivity is caused by the snow extinction properties. Conclusion would be that in L-band density affect wet snow backscattering mostly through the change of snow dielectric constant (air-snow component) and through the snow extinction properties (ground backscattering component) which makes density estimation quite complicated. Concerning depth sensitivity, by analyzing subcomponents we can see, like it was expected, that air-snow component represents a “noise”, it doesn't contain any information about the depth. Volume backscattering component also doesn't show significant depth sensitivity, which is pointing to the fact that change in albedo was primary defining density sensitivity. On the other side, ground and volume-ground interaction components are very much sensitive to depth, which is justified by the fact that their value is very sensitivity to the snow layer extinction.

(a) (b) Figure 4.5: QCA-CP model, depth sensitivity; (a) vv polarization, (b) hh polarization Even if we could estimate density and then calculate air-snow component and relatively independent volume component, we have a problem with estimating depth from the ground and volume-ground component because of their really low value.

53


Nikola BEŠIĆ

Observation concerning difference in intensity between corresponding backscattering coefficients of dry and wet snow can be now explained using difference in ground backscattering component value, which is the consequence of big wet snow absorption. IV.2.2 C band The sensitivity studies in C-band (4 – 8 GHz) are done for 6 GHz frequency value.

Figure 4.6: QCA-CP model; vv polarization Differences which can be observed relative to the L-band results are: - Significantly increased backscattering intensity, which is now, unlike it was in L-band, bigger than corresponding dry snow backscattering in the smaller angles region. - Density sensitivity is not changing like in the L-band case, now we have that backscattering is increasing all the time with density increase. - Depth sensitivity is decreased.

Figure 4.7: QCA-CP model; hh-polarization By analyzing subcomponents (figure 4.8) we have determined that cause of big rise of air-snow backscattering component, which became absolutely dominant. Ground and volume-ground backscattering components are negligible, because both of them are affected significantly by extinction, which is increased with frequency increase. It can be seen that volume backscattering component is also increased, but it is still insignificant comparing to the air-snow one. The

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dominance of the air-snow component can also explain second observation, because with density increase we have dielectric constant increase and therefore snow surface backscattering increase.

(a) (b) Figure 4.8: QCA-CP model, density sensitivity; (a) vv-polarization, (b) hh-polarization Following first observation that in the smaller angles region wet snow has a bigger backscattering in the C-band, we have compared backscattering curves of a dry snow and wet snow with the same ice volume fraction but with liquid water present (figure 4.9) and reached conclusion that this phenomena is only characteristic of the small angles region, because of the dominance of the component which comes from the smooth snow surface in case of a wet snow, while for the other angles dry snow is, just like in L band, showing more intense backscattering. This means, that like it is case with L-band, C-band can be also used for wet snow mapping, using the fact that wet snow has smaller backscattering due to the absorption, even if the wet snow surface scatters significantly.

Figure 4.9: Comparison between dry and wet snow scattering in the C-band The conclusion which can be made is that in wet snow C-band analysis we can eventualy neglect all the components except the air-snow one. Then, by using appropriate model we can deduce snow dielectric constant. The problem is how to estimate wetness and appropriate dry snow density separately, which are necessary to determine snow density. Concerning depth sensitivity, it is decreased because of the decrease of ground and ground-volume components which are mostly sensitive to depth (figure 4.10). Measurements in C-band can hardly be used for wet snow cover depth estimation.

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(a) (b) Figure 4.10: QCA-CP, depth sensitivity: (a) vv polarization, (b) hh polarization IV.2.3 X-band The sensitivity studies in X-band (8 – 12.5 GHz) are done for 10 GHz frequency value.

Figure 4.11: QCA-CP; vv polarization

Figure 4.12: QCA-CP; hh-polarization

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The most interesting facts that can be noticed are: - Density sensitivity which is approximately the same as it was in the C-band, with approximately the same value of backscattering. - Backscattering is almost totally insensitive to snow cover depth By observing subcomponents (Figure 4.13) we can notice that like in the previous transition (from L to C), extinction is increasing and therefore ground and ground-volume components are even lower than in the C-band case, while volume component is slightly increased but not enough to threaten air-snow component dominance.

(a) (b) Figure 4.13: QCA-CP model, density sensitivity: (a) vv-polarization, (b) hh-polarization The most interesting fact here is that even though we have frequency increase air-snow and thus total backscattering are not increased relative to the C-band. This can be justified by the fact that dielectric constant corresponding to certain liquid water content is lower for higher frequency, so we have compensation between those two effects. Even though volume backscattering is increased, it can be neglected, ground and volume-ground components especially, which makes measurements in X-band suitable for estimation of dielectric constant, just like it was with the C-band.

(a) (b) Figure 4.14: QCA-CP model, depth sensitivity: (a) vv-polarization, (b) hh-polarization

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Just like in the previous case, frequency increase makes components which are sensitive to depth change lower, which is leading to the fact that total backscattering is not sensitive to the depth change (figure 4.13). So, X-band measurements are not suitable for wet snow cover depth estimation.

IV.3 Multilayer wet snow cover Due to the multiple snow falling sessions during the winter we can have snow cover which is consisted out of the several layers with different properties (depth, density, wetness). Backscattering mechanism of such a snow cover is represented in the following figure (4.15). As it can be seen, because of simplicity, volume-ground interaction is neglected in the very beginning.

Figure 4.15: Multilayer backscattering mechanism A – Ground backscattering component: EM waves are reaching the ground while being refracted at the boundaries between each of the layers and attenuated by each of the layers, then it is being backscattered by the ground and again attenuated on the way back. Appropriate backscattering coefficient is given by: 𝜎𝜎𝑔𝑔0 = cos 𝜃𝜃0 𝐴𝐴𝐴𝐴𝐴𝐴𝑢𝑢𝑢𝑢 (𝑛𝑛)

𝑅𝑅(𝜃𝜃𝑛𝑛 )

cos 𝜃𝜃𝑛𝑛

𝐴𝐴𝐴𝐴𝐴𝐴𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 (𝑛𝑛),

(4.9)

where 𝐴𝐴𝐴𝐴𝐴𝐴𝑢𝑢𝑢𝑢 and 𝐴𝐴𝐴𝐴𝐴𝐴𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 are appropriate upwelling and downwelling wave attenuations, given by: 𝑘𝑘 𝑒𝑒𝑒𝑒𝑒𝑒

𝐴𝐴𝐴𝐴𝐴𝐴𝑢𝑢𝑢𝑢 (𝑘𝑘𝑒𝑒𝑒𝑒𝑒𝑒 ) = � 𝑇𝑇(𝑘𝑘−1)𝑘𝑘 exp(− 2𝜅𝜅𝑒𝑒𝑘𝑘 𝑑𝑑 𝑘𝑘 / cos 𝜃𝜃𝑘𝑘 )

(4.9a)

𝑘𝑘=1 𝑘𝑘 𝑒𝑒𝑒𝑒𝑒𝑒

𝐴𝐴𝐴𝐴𝐴𝐴𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 (𝑘𝑘𝑒𝑒𝑒𝑒𝑒𝑒 ) = � exp(− 2𝜅𝜅𝑒𝑒𝑘𝑘 𝑑𝑑 𝑘𝑘 / cos 𝜃𝜃𝑘𝑘 ) 𝑇𝑇(𝑘𝑘−1)𝑘𝑘

(4.9b)

𝑘𝑘=1

and R is the ground surface contribution, derived by using IEM. 𝑇𝑇(𝑘𝑘−1)𝑘𝑘 is Fresnel power transmission coefficient between (𝑘𝑘 − 1) and 𝑘𝑘 layer, 𝜅𝜅𝑒𝑒𝑘𝑘 - extinction coefficient of corresponding layer, 𝑑𝑑𝑘𝑘 - k layer depth and 𝜃𝜃𝑘𝑘 refracted angle in the layer. 58


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B – Volume backscattering component: EM wave are reaching the layer while being refracted at the boundaries between upper layers and attenuated by each of them, then they are scattered by the volume in the appropriate layer and again attenuated passing the same way back. Backscattering coefficient is given as: 𝑛𝑛

0 𝜎𝜎𝑣𝑣𝑣𝑣𝑣𝑣 = 4𝜋𝜋 cos 𝜃𝜃0 × � 𝐴𝐴𝐴𝐴𝐴𝐴𝑢𝑢𝑢𝑢 (𝑘𝑘 − 1)𝑇𝑇(𝑘𝑘−1)𝑘𝑘 𝑘𝑘=1

1 − exp(−2𝑘𝑘𝑒𝑒𝑘𝑘 𝑑𝑑 𝑘𝑘 / cos 𝜃𝜃𝑘𝑘 ) 𝑘𝑘 𝑃𝑃 (𝜃𝜃𝑘𝑘 )𝑇𝑇𝑘𝑘(𝑘𝑘−1) 𝐴𝐴𝐴𝐴𝐴𝐴𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 (𝑘𝑘 − 1) 2𝑘𝑘𝑒𝑒𝑘𝑘

(4.10)

C – Air – snow interface backscattering component: EM waves are being backscattered by the upper layer surface. Backscattering coefficient is given by IEM model with upper layer dielectric constant, correlation length and RMS height as input data. As we can assume according to previous chapter conclusions, we have a big problem to estimate layers depth and internal layer’s dielectric constants. The following figures are result of simulation which proves logical assumption that air-snow component which carries just upper layer properties absolutely dominates over the volume and ground component.

Figure 4.16: QCA-CP model; vv polarization; wet multilayer snow cover backscattering in L-band

Figure 4.17: QCA-CP model; vv polarization; wet multilayer snow cover backscattering in C-band

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As it is introduced through the legend, during this simulation procedure, we were varying layer densities by varying liquid water content, starting from the bottom one to the top one, with keeping all the other layers the same.

Figure 4.18: QCA-CP model; vv polarization; wet multilayer snow cover backscattering in X-band Like it can be seen, with increasing frequency, increased extinction makes ground component more and more negligible, while the volume component on the other increases. Unfortunately, even this component is not sensitive enough to internal layers densities and especially depths. It shows the biggest sensitivity for the top layer density. Multiple layer wet snow represents the biggest challenge in the snow electromagnetic modelling.

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Conclusions and Perspectives The theoretical study which is described in this report necessarily had to be done before proceeding to the further steps of our project, which is being done with the support of Electricité De France (EDF), where the data acquisition process along with “in situ” measurements must be performed. The results exposed here will be used as the acquisition guidelines. Hopefully, with some improvements which will come as a consequence of regionally characteristic input data and which will make the results more reliable because of the avoidance of ground and snow physical parameters assumptions. By applying various volume scattering models and analyzing backscattering mechanism of dry and wet snow separately, like it is only possible because they are in the electrical context totally different materials, we have reached some conclusions which will probably direct our future research which must assume work on building new semi-empirical models or expanding currently existing ones in order to use all the data available. Concerning dry snow, we have seen that the absolutely dominant backscattering component for a dry snow is the ground backscattering component, which contain density information because density affects its value using several ways, depending of frequency, but almost at all doesn’t contain depth information. In the L-band, as we have shown, density is dominantly affecting ground backscattering through the change of wave’s incident wavelength and angle, which points that the impact of all the other parameters can be suppressed by building model sensitive only to the most influential – sensor parameters. That kind of model already exists in the literature [1] and can be used for snow dielectric constant and thus density estimation. In the C-band, the ground backscattering is mostly sensitive to the change of dielectric contrast, or basically to the change of the ground dielectric properties, especially for the angles around the 40 degrees, which are standard acquisition angles. This opens space for further analysis, which must assume simulations over various ground dielectric and roughness properties before making a definite conclusion and eventually trying to build a model useful in C – band. X-band shows backscattering approximately equal sensitivity to density through both of the mentioned effects, through the change of wavelength and angle (but in opposite direction relative to the L-band), and through the change of dielectric contrast. Expanding model which already exist for L-band by using the dielectric properties of the bare ground which can be also calculated in L band [1] can be considered as the possible challenge in future. We can logically suppose that further frequency increase up to the some limits, will bring back dominance of wavelength shortening and angle change effect, but it important to notice that the ground backscattering component won’t be so dominant anymore. Depth information is mostly given through the volume backscattering component, which value is increasing with frequency, so for depth estimation lower bands are not quite useful. Model which uses C and X measurements is already built, but it can be interesting to eventually concentrate to Ku band data for this purpose. By analyzing wet snow, we have pointed out the biggest problem – the dominance of the air-snow interface backscattering component. This fact can be eventually used for successful density estimation of single layer snow cover and of top layer in case of the multilayer cover, but it 61


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represents big problem concerning depth estimation in both cases and density of the internal layers in the later case. Just like in the case of dry snow (but far less sensitive in this case), depth information is contained inside the volume backscattering component, which is being increased with frequency increase, while on the other side air-snow component doesn’t show significant rise with frequency increase above C-band, because of the wet snow dielectric constant inverse frequency dependence. So, analysis in some higher frequency bands could be done in the future as part of the effort to solve wet snow parameter estimation process. It is important to notice that all of the simulations were done for some reasonable assumed roughness and ground dielectric properties in order to achieve generalized conclusions. Therefore, it doesn’t have to mean necessary that those conclusion are applicable to any kind of snow covered terrain. For example, big change in roughness relative to the assumed one can cause “shift” of the observed behavior according to the frequency (for example, determined C-band characteristics can be noticed at lower frequencies). As it is mentioned, for EDF estimation of SWE is essentially important due to the usage of snow melting water for power plants cooling. Therefore, with their support during the next winter measurement and radar data acquisition campaign will be conducted. This will open a new space for the improvement of the snow backscattering mechanism analysis and as the direct consequence of that, for eventual semi-empirical model derivations, all in order to achieve the goal, SWE estimation using SAR remote sensing data.

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BIBLIOGRAPHY: [1] J. Shi, J. Dozier, “Estimation of Snow Water Equivalence Using SIR-C/X-SAR, Part I: Inferring Snow Density and Subsurface Properties”, IEEE Transactions on geosciences and remote sensing, vol.38, no.6, pp. 2465 – 2474, 2000 [2] J. Shi, J. Dozier, “Estimation of Snow Water Equivalence Using SIR-C/X-SAR, Part II: Inferring Snow Depth and Particle Size”, IEEE Transactions on geosciences and remote sensing, vol.38, no.6, pp. 2475 – 2488, 2000 [3] W. Gareth Rees, “Remote Sensing of Snow and Ice”, CRC Press, Taylor and Franciss group, 2006 [4] A. K. Fung, D. Schutzer, K. S. Chen, “Microwave scattering and emission models for users”, Artech House, 2010 [5] N. Longepe, “Apport de l‘imagerie SAR Sattelitaire en Bandes L et C pour la Caracterisation du Couvert Neigeux”, These de l’Universite de Rennes I, 2008 [6] N. Longepe, S. Allain, L. Ferro-Famil, E. Pottier, Y. Duran, “Snowpack Characterization in Mountainous Regions Using C-Band SAR Data and a Meteorological Model”, IEEE Transactions on geosciences and remote sensing, vol.47, no.2, pp. 406 – 418, 2009 [7] J. H. Jiang, D. L. Wu, “Ice and water permittivities for millimeter and sub-millimeter remote sensing applications”, Atmospheric Science Letters, no.5, pp. 146 – 151, 2004 [8] “Fundamentals of Remote Sensing”, Canada Centre for Remote Sensing [9] Q. Barthélemy, “Traitements des signaux RSO multidimensionnels pour l'estimation de l'équivalence en eau du manteau neigeux par télédétection radar”, Rapport de Stage Master, Grenoble INP, 2010 [10] L. Tsang, J. A. Kong, K. Ding, “Scattering of electromagnetic waves – Theories and Applications”, John Wiley & Sons, Inc. 2000 [11] L. Tsang, J. A. Kong, “Scattering of electromagnetic waves – Advanced Topics”, John Wiley & Sons, Inc. 2001 [12] C. O. Ao, “Electromagnetic Wave Scattering by Discrete Random Media with Remote Sensing Applications”, PhD Thesis, Massachusetts Institute of Technology, 2001 [13] D. Massonnet, J. C. Souyris, “Imaging witg Synthetic Aperture Radar”, EPFL Press, 2008. [14] K. P. Luojus, J. T. Pulliainen, S. J. Metsamaki, M. T. Hallikainen, “Snow-Covered Area Estimation Using Satellite Radar Wide-Swath Images”, IEEE Transactions on geosciences and remote sensing, vol. 45, no. 4, pp. 978 – 979, 2007. [15] P. Luojus, J. T. Pulliainen, S. J. Metsamaki, M. T. Hallikainen, “Accuracy Assessment of SAR Data-Based Snow-Covered Area Estimation Method”, IEEE Transactions on geosciences and remote sensing, vol. 44, no. 2, pp. 277 – 287, 2006. 63


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[16] A. N. Arslan, M. T. Hallikainen, J. T. Pulliainen, “Investigating of Snow Wetness Parameter Using a Two-Phase Backscattering Model”, IEEE Transactions on geosciences and remote sensing, vol. 43, no. 8, pp. 1827 – 1833, 2005. [17] M. T. Hallikainen, F. T. Ulaby, T. E. Van Deventer, “Extinction Behavior of Dry Snow in the 18- to 90- GHz Range”, IEEE Transactions on geosciences and remote sensing, vol. GE-25, no. 6, pp. 737 – 745, 1987. [18] M. Durand, E. J. Kim, S. A. Margulis, “Quantifying Uncertainty in Modeling Snow Microwave Radiance for a Mountain Snowpack at the Point-Scale. Including Stratigraphic Effects”, IEEE Transactions on geosciences and remote sensing, vol. 46, no. 6, pp. 1753 – 1767, 2008.

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Appendix Appendix I.1: Fresnel power coefficients and Snell’s angle Fresnel power coefficients of reflection and transmission are describing the behavior of electromagnetic (EM) wave on boundary between two different mediums in the context of power. Using those coefficients we are able to determine which part of the incident EM power is being reflected by boundary and which part is transmitted.

Figure AI.1: EM wave behavior at the boundary between two mediums If the electric field vector of the incident wave is s – polarized (perpendicular to the plane defined by incident and reflected EM wave direction vectors), Fresnel reflection coefficient is given by: 𝑛𝑛1 cos 𝜃𝜃𝑖𝑖 − 𝑛𝑛2 cos 𝜃𝜃𝑡𝑡 2 𝑅𝑅𝑠𝑠 = � � 𝑛𝑛1 cos 𝜃𝜃𝑖𝑖 + 𝑛𝑛2 cos 𝜃𝜃𝑡𝑡

(A1.1)

𝑛𝑛1 sin 𝜃𝜃𝑖𝑖 = 𝑛𝑛2 sin 𝜃𝜃𝑡𝑡

(A1.2)

where: 𝜃𝜃𝑖𝑖 is incidence angle, 𝜃𝜃𝑡𝑡 – transmitted angle, 𝑛𝑛1 - refractive index of a first medium (𝑛𝑛1 = √𝜀𝜀𝑟𝑟1 𝜇𝜇𝑟𝑟1 ), 𝑛𝑛2 - refractive index of a second medium (𝑛𝑛2 = √𝜀𝜀𝑟𝑟2 𝜇𝜇𝑟𝑟2 ). By using Snell’s law which defines directions of reflected and transmitted wave (A7), we can write (A6) as: 𝑅𝑅𝑠𝑠 =

2 𝑛𝑛 𝑛𝑛1 cos 𝜃𝜃𝑖𝑖 − 𝑛𝑛2 �1 − �𝑛𝑛1 sin 𝜃𝜃𝑖𝑖 � 2 ⎛ ⎞

2

(A1.3)

2 𝑛𝑛 𝑛𝑛1 cos 𝜃𝜃𝑖𝑖 + 𝑛𝑛2 �1 − � 1 sin 𝜃𝜃𝑖𝑖 � 𝑛𝑛2 ⎝ ⎠

Otherwise, if the electric field of the incident wave belong to the plane defined by incident and reflected EM wave direction vectors (p – polarized), the Fresnel reflection coefficient is: 𝑛𝑛1 cos 𝜃𝜃𝑡𝑡 − 𝑛𝑛2 cos 𝜃𝜃𝑖𝑖 2 𝑅𝑅𝑝𝑝 = � � = 𝑛𝑛1 cos 𝜃𝜃𝑡𝑡 + 𝑛𝑛2 cos 𝜃𝜃𝑖𝑖

2 𝑛𝑛 𝑛𝑛1 �1 − �𝑛𝑛1 sin 𝜃𝜃𝑖𝑖 � − 𝑛𝑛2 cos 𝜃𝜃𝑖𝑖 2 ⎛ ⎞ 2 𝑛𝑛 𝑛𝑛1 �1 − � 1 sin 𝜃𝜃𝑖𝑖 � + 𝑛𝑛2 cos 𝜃𝜃𝑖𝑖 𝑛𝑛2 ⎝ ⎠

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(A1.4)


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Corresponding amplitude reflection coefficients are square roots of the given power reflection coefficients, where 𝑟𝑟𝑝𝑝 = −�𝑅𝑅𝑝𝑝 , which is the consequence of the change of phase (π) during the reflection. The appropriate power transmission coefficients are defined as: 𝑇𝑇𝑠𝑠 = 1 − 𝑅𝑅𝑠𝑠 𝑇𝑇𝑝𝑝 = 1 − 𝑅𝑅𝑝𝑝

(A1.5a) (A1.5b)

Appendix I.2: Scattering matrix Scattering matrix represents mathematical description of the target in a polarimetric sense. It gives information about the change of each of the incident EM wave polarization components. Depending on several factors, it can take various forms. Factors which are defining form of the scattering matrix are: -

Degree of incident waves polarization.

If the incident waves are fully polarized, they are described by Jones vector (A1.6), consisted out of two complex elements containing information about the amplitude and phase of electric field horizontal and vertical components. 𝑗𝑗 𝜑𝜑 𝑣𝑣

𝐸𝐸 𝑒𝑒 𝐸𝐸�⃗ = � 𝑣𝑣 𝑗𝑗 𝜑𝜑 � 𝐸𝐸ℎ 𝑒𝑒 ℎ

(A1.6)

Otherwise, if the incident waves are just partially polarized, in order to describe them we are using Stokes vector (A1.7), which is consisted out of four complex elements, which are aside from components intensities and their phase difference, giving the information concerning degree of polarization. Partially polarized EM wave can be also described by Kenaugh vector, which is similar to the given Stokes one. 2

〈|𝐸𝐸𝑥𝑥 |2 + �𝐸𝐸𝑦𝑦 � 〉

-

2 ⎞ ⎛ 2 𝐼𝐼 = ⎜ 〈|𝐸𝐸𝑥𝑥 | − �𝐸𝐸𝑦𝑦 � 〉 ⎟ 2〈𝑅𝑅𝑅𝑅(𝐸𝐸𝑥𝑥 𝐸𝐸𝑦𝑦 ∗ )〉 ∗ ⎝−2〈𝐼𝐼𝐼𝐼(𝐸𝐸𝑥𝑥 𝐸𝐸𝑦𝑦 )〉⎠

(A1.7)

Backscattering coordinate system convention

In the EM wave propagation description coordinate system is placed in the way that propagation direction correspond to the z-axis, while electric and magnetic field vectors lie in the plane defined by x and y vectors. If we assume that z-axis direction remains the same after backscattering, which means that it is always positive when pointing toward the target, we can say that we using back scatter alignment (BSA) convention (figure AI.2a)

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On the other side, if z-axis direction changes after backscattering, if it is positive while pointing the target for the incident wave and positive pointing the receiver or negative pointing the target for the backscattered wave, we can say that we are using forward scatter alignment (FSA) convention (figure AI.2b).

(a) (b) Figure AI.2: (a) Back Scatter Alignment, (b) Forward scatter alignment.

So, we can now, depending on the type of the electromagnetic waves (fully or partially polarized) which are used in a remote sensing and depending on the convention which is being used, give appropriate scattering matrixes. Back Scatter Alignment Sinclair matrix (2x2) Fully polarized waves

𝑆𝑆𝑣𝑣𝑣𝑣 𝑆𝑆ℎ𝑣𝑣

𝑆𝑆𝑣𝑣ℎ � 𝑆𝑆ℎℎ

𝐾𝐾12 𝐾𝐾22 𝐾𝐾32 𝐾𝐾42

𝐾𝐾13 𝐾𝐾23 𝐾𝐾33 𝐾𝐾43

�

Kenaugh matrix (4x4)

Partially polarized waves

𝐾𝐾11 𝐾𝐾 � 21 𝐾𝐾31 𝐾𝐾41

𝐾𝐾14 𝐾𝐾24 � 𝐾𝐾34 𝐾𝐾44

Forward Scatter Alignment Jones matrix (2x2) 𝐽𝐽𝑣𝑣𝑣𝑣 𝐽𝐽ℎ𝑣𝑣

𝐽𝐽𝑣𝑣ℎ � 𝐽𝐽ℎℎ

𝑀𝑀12 𝑀𝑀22 𝑀𝑀32 𝑀𝑀42

𝑀𝑀13 𝑀𝑀23 𝑀𝑀33 𝑀𝑀43

�

Mueller matrix (4x4) 𝑀𝑀11 𝑀𝑀 � 21 𝑀𝑀31 𝑀𝑀41

𝑀𝑀14 𝑀𝑀24 � 𝑀𝑀34 𝑀𝑀44

Kenaugh and Mueller matrix, because they are referring to the intensity change, not amplitude like Sinclair and Jones one, are often called Stokes matrixes. It is important to notice that for BSA partially polarized EM wave is being described with Kenaugh, not Stokes vector. So, Sinclair and Jones matrixes are relating incident and scattered Jones vector, Kenaugh matrix relates incident and scattered Kenaugh vector, while Mueller matrix relates incident and scattered Stokes vector. In case of monostatic geometry, reciprocity is assumed (𝐻𝐻𝐻𝐻 = 𝑉𝑉𝑉𝑉) so the dimensions of the Mueller and Kenaugh matrixes are reduced to 3x3. 67


Nikola BEŠIĆ

In order to present relation between fully polarized waves in power domain, covariance matrix is used. It is derived by multiplying vector consisted of scattering matrix elements with its complex conjugated transposed version. It has 4x4 dimensions, while in case of reciprocity it is reduces to 3x3.

Appendix I.3: QCA and QCA – CP To explain properly Quasi Crystalline Approximation (QCA) and the QCA with Coherent Potentials which are used in the multiple scattering equations, we must start with single scattering treatment using Green’s functions. The single particle Green’s function satisfies following equation: �𝑗𝑗 𝐺𝐺𝑠𝑠̿ 𝐺𝐺𝑠𝑠̿ = 𝐺𝐺𝑜𝑜̿ + 𝐺𝐺𝑜𝑜̿ 𝑈𝑈

(A1.8)

𝐺𝐺𝑠𝑠̿ = 𝐺𝐺𝑜𝑜̿ + 𝐺𝐺𝑜𝑜̿ 𝑇𝑇�𝑗𝑗 𝐺𝐺𝑜𝑜̿

(A1.9)

�𝑗𝑗 + 𝑈𝑈 �𝑗𝑗 𝐺𝐺𝑜𝑜̿ 𝑇𝑇�𝑗𝑗 𝑇𝑇�𝑗𝑗 = 𝑈𝑈

(A1.10)

�𝑠𝑠 and 𝐺𝐺 �𝑜𝑜 are Green’s functions of particle and host medium, respectively, while 𝑈𝑈�𝑗𝑗 where 𝐺𝐺 represents scattering potential, which is basically difference between particle and host medium wavenumbers inside the particle, while outside has a zero value. By introducing transition operator �𝑜𝑜 = 𝑈𝑈�𝑗𝑗 𝐺𝐺 �𝑠𝑠 we can rewrite previous equation as: 𝑇𝑇�𝑗𝑗 , the way that 𝑇𝑇�𝑗𝑗 𝐺𝐺 or:

Equation A1.10 is known as Lippmann-Schwinger equation, which solution is given in a form: −1

�𝑗𝑗 𝐺𝐺𝑜𝑜̿ � 𝑈𝑈 �𝑗𝑗 𝑇𝑇�𝑗𝑗 = �𝐼𝐼 ̿ − 𝑈𝑈

(A1.11)

Having transition operator we can consider that the single particle scattering problem is solved. Now we can apply the similar principle to the multiple scattering problem. For N randomly �𝑁𝑁 , according to A1.8 is derived as: distributed particles, Green’s function 𝐺𝐺 𝑁𝑁

�𝑗𝑗 𝐺𝐺̿𝑁𝑁 𝐺𝐺̿𝑁𝑁 = 𝐺𝐺𝑜𝑜̿ + 𝐺𝐺𝑜𝑜̿ � 𝑈𝑈

(A1.12)

𝑗𝑗 =1

While for one of them, particle 𝑙𝑙, Green’s excitation function can be defined as: 𝑁𝑁

�𝑗𝑗 𝐺𝐺̿𝑁𝑁 𝐺𝐺̿𝑙𝑙 = 𝐺𝐺𝑜𝑜̿ + 𝐺𝐺𝑜𝑜̿ � 𝑈𝑈

Then, by re-writing equation A1.12 as:

(A1.13)

𝑗𝑗 =1,𝑗𝑗 ≠𝑙𝑙

�𝑗𝑗 𝐺𝐺̿𝑁𝑁 𝐺𝐺̿𝑁𝑁 = 𝐺𝐺𝑗𝑗̿ + 𝐺𝐺𝑜𝑜̿ 𝑈𝑈 68

(A1.14)


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and with introducing transition operator before, through identity - 𝑈𝑈�𝑗𝑗 𝐺𝐺̿𝑁𝑁 = 𝑇𝑇�𝑗𝑗 𝐺𝐺𝑗𝑗̿ , we can finally form system of equations called Foldy – Lax multiple scattering equations (A1.15) and by solving them for 𝐺𝐺̿𝑁𝑁 and 𝐺𝐺𝑗𝑗̿ solve multiple scattering problem numerically. 𝑁𝑁

(A1.15a)

𝑗𝑗 =1 𝑁𝑁

(A1.15b)

𝐺𝐺̿𝑁𝑁 = 𝐺𝐺𝑜𝑜̿ + 𝐺𝐺𝑜𝑜̿ � 𝑇𝑇�𝑗𝑗 𝐺𝐺𝑗𝑗̿

𝐺𝐺𝑗𝑗̿ = 𝐺𝐺𝑜𝑜̿ + 𝐺𝐺𝑜𝑜̿ � 𝑇𝑇�𝑙𝑙 𝐺𝐺̿𝑙𝑙 𝑙𝑙=1,𝑙𝑙≠𝑗𝑗

In analytical approach, we are interested in average properties from a statistical ensemble of such scatterers, so we are modifying previously given system by introducing average Green’s function. By denoting ensemble average with 𝐸𝐸, ensemble average with fixed particle 𝑗𝑗 as 𝐸𝐸𝑗𝑗 and with 𝐸𝐸𝑗𝑗𝑗𝑗 ensemble average with fixed particles 𝑗𝑗 and 𝑙𝑙, we can formulate new equations: 𝐸𝐸�𝐺𝐺̿𝑁𝑁 � = 𝐺𝐺𝑜𝑜̿ + 𝑁𝑁𝐺𝐺𝑜𝑜̿ 𝐸𝐸 �𝑇𝑇�𝑗𝑗 𝐸𝐸𝑗𝑗 �𝐺𝐺𝑗𝑗̿ ��

𝐸𝐸𝑗𝑗 �𝐺𝐺𝑗𝑗̿ � = 𝐺𝐺𝑜𝑜̿ + (𝑁𝑁 − 1)𝐺𝐺𝑜𝑜̿ 𝐸𝐸𝑗𝑗 �𝑇𝑇�𝑗𝑗 𝐸𝐸𝑗𝑗𝑗𝑗 �𝐺𝐺𝑗𝑗̿ �� , (𝑗𝑗 ≠ 𝑙𝑙)

(A1.16a)

(A1.16b)

In order to find closed solution of those equations, we must introduce some approximation. First one of them is called Quasi Crystalline Approximation. QCA assumes that conditional average holding two particles fixed is approximately equal to the conditional average holding one particle fixed. Therefore: 𝐸𝐸𝑗𝑗𝑗𝑗 �𝑄𝑄�𝑙𝑙 � = 𝐸𝐸𝑙𝑙 �𝑄𝑄�𝑙𝑙 �

(A1.17)

This way, average Green’s function can be given in a momentum representation using mass operator 𝑛𝑛0 𝐶𝐶̿ (𝑝𝑝̅ , 𝑝𝑝̅ ) [12]. 𝐺𝐺̿𝑁𝑁 (𝑝𝑝̅ ) = �𝐺𝐺𝑜𝑜̿

−1

−1

(𝑝𝑝̅ ) − 𝑛𝑛0 𝐶𝐶̿ (𝑝𝑝̅ , 𝑝𝑝̅ )�

(A1.18)

On the other side, Quasi Crystalline Approximation with Coherent Potentials is given using coherent potential operator 𝜔𝜔 �, introduced through: −1

−1 𝐺𝐺�̿0 = 𝐺𝐺𝑜𝑜̿ − 𝜔𝜔 � 1 � = 𝑈𝑈 �𝑗𝑗 − 𝜔𝜔 𝑈𝑈 � 𝑗𝑗 𝑉𝑉

� 𝑁𝑁 ) = 𝐺𝐺�� 0 , equation A1.18 has a form: so, using identity 𝐸𝐸 (𝐺𝐺 𝐺𝐺̿𝑁𝑁 (𝑝𝑝̅ ) = �𝐺𝐺𝑜𝑜̿

−1

(A1.19a)

(A1.19b)

−1

(𝑝𝑝̅ ) − 𝑛𝑛0 𝐶𝐶̂̿ (𝑝𝑝̅ , 𝑝𝑝̅ )�

(A1.20)

with the mass operator 𝑛𝑛0 𝐶𝐶̂̿ (𝑝𝑝̅ , 𝑝𝑝̅ ) depends on average Green’s function. Solving those two equation in

closed form in order to obtain average Green’s function is possible only in the low-frequency limit. Effective dielectric constants of a medium, used in the Chapters III and IV are derived that way, analytically [12]. 69


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Appendix I.4: IEM Integral equation model for backscattering is based on integrating over surface current, deduced by using surface integral, in order to get scattering field which is consequence of that current and therefore backscattering coefficient. It can be simplified by ignoring appropriate Green’s function phase, and even that way it is accurate enough to credibly present surface backscattering. Either way, here will be presented the complete model which includes Green’s function, the one which was used during the simulations. The description will be limited to the like-polarizations vv and hh. Backscattering coefficient for vertical (𝑝𝑝𝑝𝑝 = 𝑣𝑣𝑣𝑣) and horizontal (𝑝𝑝𝑝𝑝 = ℎℎ) co-polarization are given by: 0 𝜎𝜎𝑝𝑝𝑝𝑝

2 𝑘𝑘 2 σ = exp(−4k 2z σ2 ) ��(2k z σ)fpp + �Fpp 1 + Fpp 2 �� w(2k cos(θ) , 0) 4𝜋𝜋 4 ∞

n

+ � �(2k z σ) fpp n=2

2 w (n) (2k cos(θ) , 0) σ + Fpp 1 (2k z σ)n−1 � � 4 n!

(A1.21)

where: 𝜃𝜃 is the incidence angle, k – wave number, 𝜎𝜎 – RMS surface height deviation, kz - z component of wavenumber (kz = k cos(θ)), 𝑤𝑤 (𝑛𝑛) is surface spectrum, calculated as a Fourier transform of the surface correlation function, raised to n-th power: ∞

𝑤𝑤 (𝑛𝑛) (2𝑘𝑘 sin 𝜃𝜃, 0) = 2𝜋𝜋 � 𝜌𝜌𝑛𝑛 (𝑟𝑟)𝐽𝐽0 (2𝑘𝑘 sin 𝜃𝜃 𝑟𝑟)𝑟𝑟 𝑑𝑑𝑑𝑑 0

Other used functions are defined as:

𝐹𝐹𝑣𝑣𝑣𝑣1 =

𝑓𝑓𝑣𝑣𝑣𝑣 =

4𝑘𝑘

�𝜇𝜇𝑟𝑟 𝜀𝜀𝑟𝑟 − sin2 𝜃𝜃

2𝑟𝑟𝑣𝑣 2𝑟𝑟ℎ , 𝑓𝑓ℎℎ = − cos 𝜃𝜃 cos 𝜃𝜃

�(1 − 𝑟𝑟𝑣𝑣 )2 𝜀𝜀𝑟𝑟 cos 𝜃𝜃

+ (1 − 𝑟𝑟𝑣𝑣 )(1 + 𝑟𝑟𝑣𝑣 ) sin2 𝜃𝜃 ��𝜇𝜇𝑟𝑟 𝜀𝜀𝑟𝑟 − sin2 𝜃𝜃 − cos 𝜃𝜃� − (1 + 𝑟𝑟𝑣𝑣 )2 �𝜇𝜇𝑟𝑟 cos 𝜃𝜃 +

𝐹𝐹ℎℎ 1 =

−4𝑘𝑘

�𝜇𝜇𝑟𝑟 𝜀𝜀𝑟𝑟 −

sin2

𝜃𝜃

(A1.22)

+ (1 − 𝑟𝑟ℎ )(1 + 𝑟𝑟ℎ ) sin2 𝜃𝜃 ��𝜇𝜇𝑟𝑟 𝜀𝜀𝑟𝑟 − sin2 𝜃𝜃 − cos 𝜃𝜃�

70

(A1.24)

sin2 𝜃𝜃 ��𝜇𝜇𝑟𝑟 𝜀𝜀𝑟𝑟 − sin2 𝜃𝜃 − cos 𝜃𝜃�� 2𝜀𝜀𝑟𝑟

�(1 − 𝑟𝑟ℎ )2 𝜇𝜇𝑟𝑟 cos 𝜃𝜃

− (1 + 𝑟𝑟ℎ )2 �𝜀𝜀𝑟𝑟 cos 𝜃𝜃 +

(A1.23)

sin2 𝜃𝜃 ��𝜇𝜇𝑟𝑟 𝜀𝜀𝑟𝑟 − sin2 𝜃𝜃 − cos 𝜃𝜃�� 2𝜇𝜇𝑟𝑟

(A1.25)


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𝐹𝐹𝑣𝑣𝑣𝑣2 = 4𝑘𝑘 sin2 𝜃𝜃 �(1 − 𝑟𝑟𝑣𝑣 )2 �1 +

𝜀𝜀𝑟𝑟 cos 𝜃𝜃

� �𝜇𝜇𝑟𝑟 𝜀𝜀𝑟𝑟 − sin2 𝜃𝜃 cos 𝜃𝜃 − (1 − 𝑟𝑟𝑣𝑣 )(1 + 𝑟𝑟𝑣𝑣 ) �3 + � �𝜇𝜇𝑟𝑟 𝜀𝜀𝑟𝑟 − sin2 𝜃𝜃 1 𝜀𝜀𝑟𝑟 cos 𝜃𝜃 + (1 + 𝑟𝑟𝑣𝑣 )2 �1 + + �� 2𝜀𝜀𝑟𝑟 2�𝜇𝜇𝑟𝑟 𝜀𝜀𝑟𝑟 − sin2 𝜃𝜃

𝐹𝐹ℎℎ 2 = 4𝑘𝑘 sin2 𝜃𝜃 �(1 − 𝑟𝑟ℎ )2 �1 +

(A1.26)

𝜇𝜇𝑟𝑟 cos 𝜃𝜃

� �𝜇𝜇𝑟𝑟 𝜀𝜀𝑟𝑟 − sin2 𝜃𝜃 cos 𝜃𝜃 − (1 − 𝑟𝑟ℎ )(1 + 𝑟𝑟ℎ ) �3 + � �𝜇𝜇𝑟𝑟 𝜀𝜀𝑟𝑟 − sin2 𝜃𝜃 1 𝜇𝜇𝑟𝑟 cos 𝜃𝜃 + (1 + 𝑟𝑟ℎ )2 �1 + + �� 2𝜇𝜇𝑟𝑟 2�𝜇𝜇𝑟𝑟 𝜀𝜀𝑟𝑟 − sin2 𝜃𝜃

(A1.27)

where 𝜀𝜀𝑟𝑟 and 𝜇𝜇𝑟𝑟 represent, respectively, dielectric constant of the surface and its magnetic permeability, while 𝑟𝑟𝑣𝑣 and 𝑟𝑟ℎ are Fresnel amplitude reflection coefficient for vertical and horizontal polarization. Backscattering model calculated using IEM for three bands of interest for this report and with some representative input data are given in the following figure:

Figure AI.3: Backscattering curves – L band

Figure AI.4: Backscattering curves – C band 71


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Figure AI.5: Backscattering curves – X band

72


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Appendix II.1: Table 1 Coefficients used in the equation 3.26 𝑎𝑎(𝜃𝜃𝑟𝑟 , 𝑘𝑘𝑠𝑠 ) 𝑏𝑏(𝜃𝜃𝑟𝑟 , 𝑘𝑘𝑠𝑠 ) 𝑐𝑐(𝜃𝜃𝑟𝑟 , 𝑘𝑘𝑠𝑠 ) 𝑑𝑑(𝜃𝜃𝑟𝑟 , 𝑘𝑘𝑠𝑠 ) -1.398 -2.580 -1.183 -1.999 0.134 2.840 0.530 0.942 0.126 -6.980 0.000 -0.464 -0.128 3.910 -0.174 0.000 -23.694 -20.668 -22.822 -22.943 112.035 109.549 111.070 112.002 -264.458 -257.470 -261.271 -263.821 283.381 274.670 278.709 281.344 -110.814 -107.743 -108.116 -108.644 0.000 0.234 0.000 0.000 -2.416 -2.437 -2.437 -2.472 Table AII.1 : Coefficients used in equation 3.26

A0 A1 A2 A3 B0 B1 B2 B3 B4 L1 S1

𝑒𝑒(𝜃𝜃𝑟𝑟 , 𝑘𝑘𝑠𝑠 ) -3.333 0.000 1.200 -0.787 -21.941 110.716 -260.213 277.424 -108.445 0.000 -2.344

Appendix II.2: Tables 2 and 3 Coefficients used in the equation 3.29

C-VV k 1 s