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**Radiative Transfer Dr. X-Pol Microwave Remote Sensing INEL 6669**

Dept. of Electrical & Computer Engineering, UPRM, Mayagüez, PR INEL 6069,

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**Outline Theory of Radiative Transfer TAP of absorbing/scattering Media**

Extinction &Emission Equation of Transfer TAP of absorbing/scattering Media TAP of atmosphere & Terrain: upwelling and downwelling Emission and Scattering by Terrain Homogeneous terrain medium with uniform T profile non-uniform T&er profile: coherent & incoherent approach Emissivity of dielectric Slab Emissivity of Rough surface INEL 6069,

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**Radiative Transfer Some energy is transmitted**

some is scattered, some is absorbed INEL 6069,

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**Outline Theory of Radiative Transfer TAP of absorbing/scattering Media**

Extinction &Emission Equation of Transfer TAP of absorbing/scattering Media TAP of atmosphere & Terrain: upwelling-down-welling Emission and Scattering by Terrain Homogeneous terrain medium with uniform T profile non-uniform T&er profile: coherent & incoherent approach Emissivity of dielectric Slab Emissivity of Rough surface INEL 6069,

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Radiative Transfer, Radiative Transfer :Extinction Interaction between radiation and matter Extinction (Change due to energy loss) dIextin= ke I(r,r’)dr two processes: absorbtion & scattering ke= ka + ks in Nepers/m where ke is the extinction or power attenuation coefficient, it’s due to absorption and scattering away in other direction. dA dr Volume of some matter Incident radiation exiting INEL 6069,

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**Radiative Transfer : Emission Interaction between radiation and matter**

Emission (Change due to energy gained) dIemit= (kaJa + ksJs )dr two processes or mechanisms: emit & scatter [Ja & Js account for thermal emission & scattering] Introduce the scattering albedo, a =ks/ke dIemit= [(ke - ks)Ja + ks Js )] dIemit= ke [(1- a)Ja + aJs )]= keJdr where J = (1- a)Ja + aJs dA dr Js Ja INEL 6069,

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**albedo http://www.sciencedaily.com/releases/2014/02/140219115110.htm**

The retreat of sea ice in the Arctic Ocean is diminishing Earth's albedo, or reflectivity, by an amount considerably larger than previously estimated, according to a new study that uses data from instruments that fly aboard several NASA satellites. The study, conducted by researchers at Scripps Institution of Oceanography, at the University of California, San Diego, uses data from the Clouds and Earth's Radiant Energy System, or CERES, instrument. There are CERES instruments aboard NASA's Tropical Rainfall Measurement Mission, or TRMM, satellite, Terra, Aqua and NASA-NOAA's Suomi National Polar-orbiting Partnership (Suomi NPP) satellites. The first CERES instrument was launched in December of 1997 aboard TRMM. As the sea ice melts, its white reflective surface is replaced by a relatively dark ocean surface. This diminishes the amount of sunlight being reflected back to space, causing Earth to absorb an increasing amount of solar energy. The Arctic has warmed by 3.6 F (2 C) since the 1970s. albedo

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**Outline Theory of Radiative Transfer TAP of absorbing/scattering Media**

Extinction &Emission Equation of Transfer TAP of absorbing/scattering Media TAP of atmosphere & Terrain: upwelling-down-welling Emission and Scattering by Terrain Homogeneous terrain medium with uniform T profile non-uniform T&er profile: coherent & incoherent approach Emissivity of dielectric Slab Emissivity of Rough surface INEL 6069,

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**Radiative Transfer : Equation of Transfer I**

dI= I(r+dr) - I(r) = dIemit -dIextin =ke (J-I)dr = (J-I)dt where dt = ke dr is the optical depth, therefore, and t is the optical thickness or opacity in Np. dA dr B(r) B(r+dr) INEL 6069,

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**Radiative Transfer : Equation of Transfer II**

dI=(J-I)dt or dI/dt + I = J Multiplying by et(0,r’) and integrating from 0 to r. Horizontal layers (stratified atm) B(r) r r’ B(0) INEL 6069,

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**Outline Theory of Radiative Transfer TAP of absorbing/scattering Media**

Extinction &Emission Equation of Transfer TAP of absorbing/scattering Media TAP of atmosphere & Terrain: upwelling-down-welling Emission and Scattering by Terrain Homogeneous terrain medium with uniform T profile non-uniform T&er profile: coherent & incoherent approach Emissivity of dielectric Slab Emissivity of Rough surface INEL 6069,

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**TAP of absorbing/scattering Media (1)**

Radiative Transfer, TAP of absorbing/scattering Media (1) In the microwave region, where R-J applies, there’s a T to I Similarly, under thermodynamic equilibrium (emission = absorption) the absorption source function is [Recall J = (1- a) Ja + aJs ] where T is the physical temperature of the medium. INEL 6069,

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**TAP of absorbing/scattering Media (2)**

Radiative Transfer, TAP of absorbing/scattering Media (2) Similarly, for the scattering source function, Js where Y is the phase function and accounts for the portion of incident radiation scattered from direction ri into direction r, where we have defined a scattered radiometric temperature as, In eq Ulaby & Long, the multiply by ½ INEL 6069,

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**TAP of absorbing/scattering Media (3)**

Radiative Transfer, TAP of absorbing/scattering Media (3) In terms of temperature, ke= ka + ks For scatter-free medium: ks(= ake)=0 then a= 0 and ke= ka and the opacity is INEL 6069,

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**Brightness temperature**

Define the 1-way atmospheric transmissivity:

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**TAP of absorbing/scattering Media III**

Radiative Transfer, TAP of absorbing/scattering Media III Ex. For scatter-free medium: An airborne radiometer measuring ice. TAP (0)= Tice e-t (0,H) = attenuation of atmosphere up to height H INEL 6069,

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**Scattering Rain and clouds produce a bit @ microwaves.**

Radiative Transfer, Scattering Rain and clouds produce a microwaves. Can be neglected for f under 10 GHz. Surface scattering - depends on the interface, dielectric properties, geometry. Volume scattering- occurs for l ~dia & dist. l>>d appears homogeneous INEL 6069,

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**Outline Theory of Radiative Transfer TAP of absorbing/scattering Media**

Extinction &Emission Equation of Transfer TAP of absorbing/scattering Media TAP of atmosphere & Terrain: upwelling-down-welling Emission and Scattering by Terrain Homogeneous terrain medium with uniform T profile non-uniform T&er profile: coherent & incoherent approach Emissivity of dielectric Slab Emissivity of Rough surface INEL 6069,

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**TAP of atmosphere & Terrain: Upwelling**

Radiative Transfer, TAP of atmosphere & Terrain: Upwelling r’=z’secq Upwelling (no ground) All the upward radiation emitted by the entire atmospheric path between the ground and the observation point. If H>20km q No ground contribution INEL 6069,

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**TAP of atmosphere & Terrain: Downwelling**

Radiative Transfer, TAP of atmosphere & Terrain: Downwelling Downwelling r’=z’secq INEL 6069,

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**TAP of atmosphere & Terrain: downwelling**

Radiative Transfer, TAP of atmosphere & Terrain: downwelling Special case: plane homogeneous atmosphere (or cloud) with T(z)=To and ka=kao over the range z=0 to z=H. *Used in programming codes. INEL 6069,

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**TAP of Atmosphere & Terrain**

Radiative Transfer, TAP of Atmosphere & Terrain Upwelling Radiation (with ground) Where TB(0) is the contribution from the surface emissions and reflections (from downwelling and cosmic radiation) which is treated in more detail in the following chapter. INEL 6069,

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**Brightness Temperature**

Radiative Transfer, Brightness Temperature Self-emitted radiation from the surface (e.g. terrain, ice, ocean) upward radiation from the atmosphere downward-emitted atmospheric radiation that is reflected by the surface in the antenna direction virtually no solar contamination INEL 6069,

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**Radiative Transfer Theory**

Interaction between radiation and matter : processes => emission & extinction (s & a) Under clear sky conditions - no scattering INEL 6069,

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**Example: Upward-looking radiometer like Arecibo looks at...**

Radiative Transfer, Example: Upward-looking radiometer like Arecibo looks at... Brightness Temperature [K] Frequency [GHz] where TC is the cosmic radiation INEL 6069,

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**Outline Theory of Radiative Transfer TAP of absorbing/scattering Media**

Extinction &Emission Equation of Transfer TAP of absorbing/scattering Media TAP of atmosphere & Terrain: upwelling-down-welling Emission and Scattering by Terrain Homogeneous terrain medium with uniform T profile non-uniform T&er profile: coherent & incoherent approach Emissivity of dielectric Slab Emissivity of Rough surface INEL 6069,

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**Flat surface vs. rough surface**

Radiative Transfer, Emission and Scattering by Terrain: relate TB & TSC to the medium properties. Flat surface vs. rough surface Flat surface (Specular surface) h<<l Height variations are much smaller than wavelength of radiation. Snell’s Law applies. Rough surface hl Lambertian surface is considered “perfectly” rough. Many times is a combination of both. INEL 6069,

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**Properties of the Specular Surface**

Radiative Transfer, Properties of the Specular Surface ec2 mr2 ec1 =e’ -je” mr1 q1 q2 Fresnel reflection gives the power reflectivity where, The power transmissivity is, and, INEL 6069,

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**Properties of the Specular Surface**

Radiative Transfer, Properties of the Specular Surface e2 m2 ka2 ec1 mr1 q1 q2 Snell’s Law relates the angles of the incidence and transmission. The field attenuation coefficient is [Np/m] The power attenuation coefficient is ka= 2a [Np/m] INEL 6069,

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**Homogeneous terrain medium; Assuming uniform T profile, T(z) = Tg**

Radiative Transfer, Homogeneous terrain medium; Assuming uniform T profile, T(z) = Tg Snell’s Law: e2 m2 Ka2=2a2 ec1 mr1 T(z) =Tg TSC TB TDN The brightness temperature is The upwelling temperature of homogeneous terrain is The scattered temperature is The emissivity of such isothermal medium is e(q;p) =TB/Tg = 1-G1 INEL 6069,

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**TB transmission across Specular boundary**

Radiative Transfer, TB transmission across Specular boundary Snell’s Law: (1) Differentiating wrt q and multiplying by df on both sides (2) Multiplying (1) and (2) INEL 6069,

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**TB transmission for Specular**

Radiative Transfer, TB transmission for Specular Dividing P1/P2: This is the transmissivity INEL 6069,

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**TB transmission for Specular**

Radiative Transfer, TB transmission for Specular Since power is proportional to TB: Where the reflectivities (ch.2): INEL 6069,

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**Emissivity at 10GHz for specular surface for H and V polarizations**

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**Homogeneous terrain medium**

Radiative Transfer, Homogeneous terrain medium The apparent temperature from specular surface is then Emission by the ground Reflections from the atmosphere INEL 6069,

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Example

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Radiative Transfer, Probl. 4.5 where L=et INEL 6069,

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**Assigned Problems Ulaby & Long 2013 6.1-7, 6.10, and 12**

Exam next wed mar 5

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**Emissivity of Rough surface**

Radiative Transfer, Emissivity of Rough surface When the surface is rough in terms of wavelength, there are small height irregularities on the surface that scatter power in many directions. There will be an angular dependency. Pattern of radiation transmitted across the boundary from the thermal radiation incoming from the medium. air medium INEL 6069,

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**Rough surface emissivity**

Medium (b) with small irregularities on the order of the wavelength. Part of the scatter power is reflected in specular direction and it’s mostly phase-coherent. The remainder is diffuse or phase-incoherent. Part is the same polarization as incident Rest is orthogonal pol Law of thermodynamic equilibrium requires absorptivity=emissivity:

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**Rough surface emissivity**

For wave incident in medium 1 upon medium 2, the power absorbed by medium 2 air medium Same polarization as incident orthogonal polarization Where the scattered fields at a distance Rr are related to the incident fields by:

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**Rough surface emissivity**

Find Reflectivity Substitute and get emissivity = 1-reflectivitty FSA= forward scattering alignment where Spq are FSA scattering amplitudes & we used the backscattering coefficient or RCS defined by: Then the scattered power is given by:

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**Rough surface emissivity**

Find Reflectivity Substitute and get emissivity = 1-reflectivitty Where the backscattering coefficient consists of coherent component along the specular direction and incoherent component along all directions. (Chapter 5)

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**Rough surface emissivity**

Substitute and get emissivity = 1-reflectivitty Where the coherent component is given by: (Chapter 5) where y is the roughness parameter, given by , s=rms height

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**Rough surface emissivity**

The eq. for Gh can also be used to relate surface scattered brightness temperature, TSS to the unpolarized emitted atmospheric temperature TDN coming down from all directions in the upper hemisphere. For h pol: Accounts for the incoherent part of the scattering by the surface Similar for v pol

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**Lambertian surface emissivity**

For Lambertian surface (perfectly rough): Which is polarization and angle independent, and so is a constant related to the permittivity of the surface.

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**Emissivity of 2-layer Composite**

In thermodynamic equilibrium we have: Example: a 20GHz nadir looking radiometer maps the thickness, d, of oil spill over ocean at To=293K. Plot increase in TB as function of oil thickness. Where the emissivity is related to the reflectivity as (for v or h): Ch.2 Both medium 2 and 3 are lossy.

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Online modules

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Radiative Transfer, *subroutine to compute Tdownwelling where ka= water + oxygen attenuations * Begin computation of radiative transfer integral *Integration of Tb_dn, Tb_up, opacity DO 45 ifr = 1 ,3 freq =fre(ifr) ** Compute the DOWNWELLING BRIGHTNESS TEMPERATURE ********* TBdn(ifr) = 0. opa = 0.0 TAU = 1.0 DO 130 J = 1, JJ vap =v(j) pre =p(j) tem =t(j) absorp=VAPOR(freq,Vap,Pre,Tem,CL,CW,CC)+ OXYGEN(freq,Vap,Pre,Tem,CX) LTMP0 = EXP(-DZ*absorp) DTB = Tem*(1.0 - LTMP0)*TAU TAU = TAU*LTMP0 TBdn(ifr) = TBdn(ifr) + DTB 130 CONTINUE tauf(ifr) = tau * Add on freq dependent cosmic background term: TBdn(ifr) = TBdn(ifr) + ( *(freq-18))*TAU * *COMPUTE THE UpWELLING BRIGHTNESS TEMPERATURE*********** TBup(ifr) = 0. DO 135 J = JJ,1,-1 LTMP0 = EXP(-DZ*(VAPOR(freq,Vap,Pre,Tem,CL,CW,CC)+ OXYGEN(freq,Vap,Pre,Tem,CX)) ) TBup(ifr) = TBup(ifr) + DTB CONTINUE 45 CONTINUE INEL 6069,

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Radiative Transfer, Code for Tdn ** Compute the DOWNWELLING BRIGHTNESS TEMPERATURE ********* TBdn = 0. TAU = 1.0 jj=1000 dZ=30000/jj DO 130 J = 1, JJ vap =v(j); pre =p(j); tem =t(j) Ka=VAPOR(freq,Vap,Pre,Tem,CL,CW,CC)+ OXYGEN(freq,Vap,Pre,Tem,CX) LTMP0 = EXP(-dZ*Ka) DTB = Tem*(1.0 - LTMP0)*TAU TAU = TAU*LTMP0 TBdn = TBdn + DTB 130 CONTINUE * Add on freq dependent cosmic background term: TBdn(ifr) = TBdn(ifr) + ( *(freq-18))*TAU INEL 6069,

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**Code for Tup *COMPUTE THE UpWELLING BRIGHTNESS TEMPERATURE*************

Radiative Transfer, Code for Tup *COMPUTE THE UpWELLING BRIGHTNESS TEMPERATURE*********** TBup(ifr) = 0. TAU = 1.0 DO 135 J = JJ,1,-1 vap =v(j) pre =p(j) tem =t(j) LTMP0 = EXP(-DZ*ka(j)) DTB = Tem*(1.0 - LTMP0)*TAU TAU = TAU*LTMP0 TBup(ifr) = TBup(ifr) + DTB CONTINUE 45 CONTINUE INEL 6069,

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