1. Radiative transfer with emission. Atmosphere and surfaces emit infrared and microwave radiation. Recall the Beer-Bouguer-Lambert law for emission

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1 Lecture 8. Princiles of assive remote sensing using emission an alications: Remote sensing of atmosheric ath-integrate quantities clou liqui water content an reciitable water vaor.. Raiative transfer with emission.. Microwave raiative transfer. 3. Measurements of atmosheric ath-integrate quantities reciitable water clou liqui water. Require reaing: S: Petty: 8 Aitional reaing: Microwave remote sensing various satellite ata roucts incluing SSM/: htt:// Examle: NOAA AMSU-A retrieval algorithm for total reciitable water an clou liqui water: htt:// Raiative transfer with emission. Atmoshere an surfaces emit infrare an microwave raiation. Accoring to the Kirchhoff s law: emission=absortion Recall the eer-ouguer-lambert law for emission ke J s where k e is the volume extinction coefficient along ath s. For a non-scattering meium in LE the Planck function gives the source function J [8.]

2 Neglecting scattering => volume extinction coefficient = volume absortion coefficient hus the net change of raiation along ath s is ue to the combination of emission an extinction = extinction + emission an thus the raiative transfer equation in the thermal region ignoring scattering is or s k s k s [8.] k ] [8.3] [ NOE: Eqs.[8.]-[8.3] are often calle the ifferential forms of the raiative transfer equation. Recall that by efinition k s s Let s re-arrange terms in Eq.[8.3] an multily both sies by ex- ex ex ex [8.4] an using that [xex-x]=ex-xx - ex-xxx we have ex ex [8.5] ntegrating the above equation along a ath extening from some oint s to the en oint s it becomes s s s s s e s e s e s [8.6] s an re-arranging terms we have the solution of the raiative transfer in R s [ s s] [ s s ] s s e s e s [8.7] s contribution from raiation incient at s an transmitte to s contribution from raiation emitte along the ath an transmitte to s

3 Let s consier a lane-arallel atmoshere z= s an z = s Uwar intensity is for Downwar intensity is for or / or using that cos= cos/= an cos =- / z = z to z z = bottom = NOE: For ownwar intensity is relace by. Eq.[8.7] gives both the uwar intensity in the lane-arallel atmoshere ex [8.8] ex an the ownwar intensity in the lane-arallel atmoshere: ex ex [8.9] 3

4 4 n the atmosheric conitions for R raiation one can consier that at the surface s or s no thermal incient raiation at the OA no eenence on azimuthal angle. hus Eqs.[8.8] an [8.9] can be re-written as in the wavenumber omain ex ex [8.] ex [8.] Eqs.[8.] - [8.] can be exresse in terns of monochromatic transmittance. Recall that ex [8.] an the ifferential form is ex [8.3] Multilication law of transmittance states that when several gases absorb the monochromatic transmittance is a rouct of the monochromatic transmittances of iniviual gases: N N [8.4]

5 5 hus the general solutions for monochromatic uwar an ownwar raiances in terms of transmittance are: [8.5] [8.6] he general solutions for monochromatic uwar an ownwar raiances can be exresse in terms of the weighing function: W ' [8.7] where the weighting function is efine as W ' ' [8.8] NOE: he concet of the weighting function lays a central role in souning techniques i.e. retrievals of the vertical rofile of an gas HO O3 concentrations from hyersectral assive remote sensing. Consier an isothermal atmoshere. hen =z = constant= atm he raiance at the to of the isothermal atmoshere from Eq.[8.5] is atm [8.9] Where is the transmission function of the entire atmoshere. n no scattering then Emissivity = Absortivity = - ransmission hus Eq.[8.9] can be re-written in terms of emissivity or absortivity.

6 . Microwave raiative transfer. Accoring to the Rayleigh-Jeans istribution see Lecture : rightness temerature is linear roortional to the raiance n the microwave surface emissivities are low => nee to account for reflection i.e. the ortion of microwave raiation emitte by the atmoshere towar the ocean is reflecte back to the atmoshere an can be olarize eening on the viewing irection. Eq.[8.8] can be moifie to give the brightness temerature assive microwave etector at a wavenumber b measure by a satellite where b R sur ex / ex / atm atm ex / / ex / / [8.] sur is the surface temerature atm is the atmosheric temerature is the emissivity of the ocean surface with the given olarization state an R is reflectivity of the ocean surface with the given olarization state. Let s assume that the absortion by water vaor only in the bounary layer ex / / [ ex / ] [8.] atm hus we have from Eq.[.] b sur [ ] [8.] where ex is the transmission function. sur 6

7 3. Measurements of atmosheric ath-integrate quantities: reciitable water vaor an clou liqui water. Let s consier brightness temerature measure at 9.35 GHz an 37 GHz for two measure olarization state horizontal H an vertical V olarization states Using Eq.[8.] we have at each frequency where R H V ~ ~ H V b ~ sur V H R R ~ [8.3] he atmosheric transmission can be reresente as a combination of transmission for O O clou liqui water W an water vaor at each frequency [8.4] O W W ex k LWP / where a W is the mass absortion coefficient of liqui water a W k clou ros an LWP is the liqui water ath efine as liqui water content LWC Lecture 4 Eq.[4.33] integrate over the ath. ex k / where k is the absortion coefficient of water vaor an is the amount of water vaor integrate over the art calle reciitable water often reorte in mm. From Eq.[8.3] we have k a W b LWP k ln[ ] [8.5] V H R R sur Eq.[8.5] for two channels => we have two equations to solve for LWP an given the values of O9 O37 k a W k an H V R ~ Problems: Nee to know absortion coefficients H V R ~ are functions of win see 7

8 Examle: he above rincile rovie a basis for the retrieval algorithm of Secial Sensor Microwave/mager SSM/. SSM/ is a assive microwave sensor aboar the DMSP satellite series. he SSM/ time series consists of 6 satellites covering the erio from 987 to the resent. htt:// Examle of SSM/ roucts Daily UC AM retrievals of reciitable water vaor Weekly retrievals of reciitable water vaor 8

9 Daily UC AM retrievals of clou liqui water - comare to the above image for reciitable water Weekly retrievals of clou liqui water - comare to the above image for reciitable water 9