=Paper= {{Paper |id=Vol-2534/72_poster_paper |storemode=property |title=Sensitivity of Outgoing Longwave Radiation to Variations of Underlying Surface Emissivity Coefficient |pdfUrl=https://ceur-ws.org/Vol-2534/72_poster_paper.pdf |volume=Vol-2534 |authors=Egor Yu. Mordvin,Anatoly A. Lagutin,Rimma V. Kravchenko }} ==Sensitivity of Outgoing Longwave Radiation to Variations of Underlying Surface Emissivity Coefficient== https://ceur-ws.org/Vol-2534/72_poster_paper.pdf 
        Sensitivity of Outgoing Longwave Radiation to Variations of
                  Underlying Surface Emissivity Coefficient

                           Egor Yu. Mordvin, Anatoly A. Lagutin, Rimma V. Kravchenko

                                        Altai State University, Barnaul, Russia




              Abstract. We study the sensitivity of outgoing longwave radiation to variations of
              underlying surface emissivity πœ€Ρ (𝜈). The approach to solving the problem under
              consideration is based on the functional theory of sensitivity. Using the analytical result
              obtained in the work on the differential sensitivity coefficient of the outgoing radiation as
              well as the computational complex created on the basis of the LBLRTM model, it is
              shown that the maximum sensitivity of the outgoing longwave radiation to the variations
              of πœ€Ρ (𝜈) is observed in the ranges of 780–1000 cm-1 and 1015–1200 cm-1.

              Keywords: outgoing longwave radiation, sensitivity, emissivity, underlying surface,
              LBLRTM.

1        Introduction
    Outgoing longwave radiation is one of the key components of the Earth radiation balance [1,2]. It characterizes
the amount of energy generated by the radiation of the underlying surface of the Earth and the ascending radiation of
the atmosphere, which goes into space from the β€œunderlying surface – atmosphere” system.
    Let the temperature, pressure and the emissivity of the underlying surface be denoted as 𝑇𝑠 , 𝑃𝑠 and πœ€π‘  (𝜈), the
Planck function will be denoted as 𝐡(𝜈, 𝑇𝑠 ), and by 𝜏(𝜈, 𝑃 β†’ 0; πœƒ) we shall mean transmission function of the
atmospheric radiation with the frequency 𝜈 on the path β€œthe atmospheric level with pressure 𝑃 – satellite”. In this case
the spectral intensity 𝐿𝐢𝐿𝑅 (𝜈, πœƒ) of the outgoing from the cloudless nonscattering atmosphere under a zenith angle πœƒ
radiation can be represented as [3]
                                                                        0
                                                                                 π‘‘πœ(𝜈, 𝑃 β†’ 0; πœƒ)
               𝐿𝐢𝐿𝑅 (𝜈, πœƒ) = πœ€π‘  (𝜈)𝐡(𝜈, 𝑇𝑠 )𝜏(𝜈, 𝑃𝑠 β†’ 0; πœƒ) + ∫ 𝐡(𝜈, 𝑇(𝑃))                       𝑑 ln(𝑃).            (1)
                                                                       𝑃𝑠           𝑑 ln(𝑃)
    It should be noted that in this work we neglect the contributions of solar radiation and the rescattering of
descending radiation by underlying surface into the spectral intensity (1) of outgoing longwave radiation (OLR).
    It can also be shown that if the fraction 𝛼 of a pixel is covered by cloud at pressure level 𝑃с , whose emissivity is
πœ€Ρ (𝜈) and the temperature at the upper edge is 𝑇с , then the spectral intensity of the radiation leaving the cloudy
atmosphere is described by the expression
                                  𝐿(𝜈, πœƒ) = (1 βˆ’ π›Όπœ€π‘ (𝜈))𝐿𝐢𝐿𝑅 + π›Όπœ€π‘ (𝜈)𝐿𝐢𝐿𝐷 (𝑃с ),                                   (2)
where
                                                                   0
                                                                               π‘‘πœ(𝜈, 𝑃 β†’ 0; πœƒ)
                    𝐿𝐢𝐿𝐷 (𝑃с ) = 𝐡(𝜈, 𝑇𝑐 )𝜏(𝜈, 𝑃𝑐 β†’ 0; πœƒ) + ∫ 𝐡(𝜈, 𝑇(𝑃))                       𝑑 ln(𝑃)
                                                                  𝑃𝑐              𝑑 ln(𝑃)
is the intensity of radiation emanating from an opaque cloud at cloud top pressure 𝑃с .
    The flux of the outgoing longwave radiation 𝐹 was found by integrating the intensity (2) with respect to angles
and frequency. In the case of azimuthal symmetry of the outgoing radiation, the OLR flux is determined by the
equation
                                                 πœ‹β„2          ∞
                                      𝐹 = 2πœ‹ ∫         π‘‘πœƒ ∫ 𝐿(𝜈, πœƒ) sin πœƒ cos πœƒ π‘‘πœˆ .
                                                0           0
    In this paper, the integral of the spectral OLR with respect to angle πœƒ was calculated in the framework of the
β€œeffective angle approximation”, which is determined by the expression [4,5]
                                                        πœ‹β„2
                             𝐹𝜈 = 2πœ‹πΏ (𝜈, πœƒπ‘’π‘“ (𝜈)) ∫          sin πœƒ cos πœƒ π‘‘πœƒ = πœ‹πΏ (𝜈, πœƒπ‘’π‘“ (𝜈)).
                                                        0
    Using this approach for the flux 𝐹, we obtain:


Copyright Β© 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0
International (CC BY 4.0).
                                                       ∞
                                               𝐹 = πœ‹ ∫ 𝐿 (𝜈, πœƒπ‘’π‘“ (𝜈)) π‘‘πœˆ .                                           (3)
                                                      0
    For the range 2–2750 cm-1, the values of effective angles obtained in [4] are given in table 1.

                           Table 1. Dependence of the effective angle πœƒπ‘’π‘“ on frequency [4].

                    βˆ†πœˆ, cm-1    πœƒπ‘’π‘“ , degree     βˆ†πœˆ, cm-1    πœƒπ‘’π‘“ , degree     βˆ†πœˆ, cm-1     πœƒπ‘’π‘“ , degree
                     2–150           50.36       650–800        52.94        1500–1800        50.36
                    150–250          50.64       800–950        55.86        1800–2100        52.51
                    250–350          51.62      950–1100        53.08        2100–2400        52.74
                    350–500          52.57      1100–1250       55.21        2400–2750        55.08
                    500–650          52.53      1250–1500       51.64

    The equations (2)–(3) show that the OLR flux is determined by the emissivity and the temperature of the
underlying surface, temperature and humidity profiles, cloud properties, as well as concentrations of greenhouse
gases and aerosols in the atmosphere. Due to the presence of nonlinear relationships between these characteristics of
the system and the OLR, the interpretation of experimental results obtained by satellite devices requires data on the
OLR sensitivity to system characteristic variations.
    To solve such problems, in our work [6], we propose an approach based on the functional theory of sensitivity [7].
The coefficients of the differential sensitivity of the infrared spaceborne hyperspectrometer’s readings to variations of
the gas composition of the atmosphere have been obtained. It is shown that this coefficient is expressed in terms of
the mass absorption coefficient of the studied gas and the universal function determined by the intensity of the
outgoing radiation for the undisturbed atmosphere.
    In this paper, this approach is used to analyze the effect of variations in the emissivity on the flux of longwave
radiation leaving the atmosphere. The relevance of this study is due to the need to estimate how the OLR is affected
by the changes in the structure of underlying surface caused by both climate changes in global and regional scales and
landuse, to take into account errors in the dependence of the emissivity on the wavenumber during the interpretation
of satellite data, and to verify the correctness of the πœ€π‘  (𝜈) definition in climate models.

2        The sensitivity of the outgoing radiation flux to variations of the underlying surface
   Following [6], the variation of the flux π›₯𝐹,
                                        π›₯𝐹(𝜈(βˆ™) β†’ 𝜈 β€² (βˆ™)) = 𝐹(𝜈 β€² (βˆ™)) βˆ’ 𝐹(𝜈(βˆ™)),
which is due to change of emissivity πœ€π‘  (𝜈) β†’ πœ€π‘ β€² (𝜈) = πœ€π‘  (𝜈) + π›₯πœ€π‘  (𝜈), will be presented in the form
                                                    ∞
                                                       𝛿𝐹(πœ€π‘  (βˆ™))
                                           π›₯𝐹 = ∫                   π›₯πœ€π‘  (𝜈0 )π‘‘πœˆ0 .                                   (4)
                                                   0  π›Ώπœ€ 𝑠 (𝜈0 )π‘‘πœˆ0

   The first functional derivative in (4) is conventionally called the differential sensitivity coefficient (see [6,7]).
Function
                                                   πœ€π‘  (𝜈0 ) 𝛿𝐹(πœ€π‘  (βˆ™))
                                               𝑆=
                                                      𝐹 π›Ώπœ€π‘  (𝜈0 )π‘‘πœˆ0
describes a percentage variation 𝐹 caused by a change of πœ€π‘  in a unit interval around 𝜈0 by 1%.
   Calculating the variational derivative, we find the coefficient of differential sensitivity and the variation of the
OLR flux:
                            𝛿𝐹(πœ€πœˆ (βˆ™))
                                        = πœ‹(1 βˆ’ π›Όπœ€π‘ (𝜈0 ))𝐡(𝜈0 , 𝑇𝑠 )𝜏 (𝜈0 , 𝑃𝑠 β†’ 0; πœƒπ‘’π‘“ (𝜈0 )),                    (5)
                           π›Ώπœ€π‘  (𝜈0 )π‘‘πœˆ0
                                  ∞
                        π›₯𝐹 = πœ‹ ∫ (1 βˆ’ π›Όπœ€π‘ (𝜈0 ))𝐡(𝜈0 , 𝑇𝑠 )𝜏 (𝜈0 , 𝑃𝑠 β†’ 0; πœƒπ‘’π‘“ (𝜈0 )) π›₯πœ€πœˆ (𝜈0 )π‘‘πœˆ0 .                 (6)
                                 0


3        Results
   Calculations of the atmospheric transmission function 𝜏(𝜈, 𝑃𝑠 β†’ 0; πœƒπ‘’π‘“ ), the OLR flux and the differential
sensitivity coefficient were performed using the LBLRTM (Line-By-Line Radioactive Transfer Model) [8]. Preparing
the necessary characteristics of the atmosphere and the underlying surface for the model as well as generating a
configuration file for LBLRTM run was carried out using the program developed by the authors. Atmospheric and
surface data were extracted from the research product AIRS version 6, which contains measurements at 100
atmospheric levels with the altitude range from the surface up to ~60 km. To set the emissivity of the underlying
surface we use data from the MODIS UCSB Emissivity Library of the MODIS LST group at University of California,
Santa Barbara (UCSB) (http://www.icess.ucsb.edu/modis/EMIS/html/em.html).
    Figures 1 and 2 show the dependence of flux sensitivity 𝑆π›₯𝜈 on frequency in a cloudless atmosphere for a sandy
surface and for a surface covered with vegetation. It is easy to see that the sensitivity peaks are in the ranges of 780–
1000 cm-1 and 1015–1200 cm-1. Significantly lower sensitivity is observed in the ranges 2100–2200 cm-1 and 2480–
2550 cm-1.




Figure 1. The sensitivity of the flux 𝑆π›₯𝜈 to the variation of the emissivity πœ€πœˆ by 1% in the interval π›₯𝜈 = 𝜈 ⁄1200 for
                                                   a sandy surface.




Figure 2. The sensitivity of the flux 𝑆π›₯𝜈 to the variation of the emissivity πœ€πœˆ by 1% in the interval π›₯𝜈 = 𝜈 ⁄1200 for
                                          a surface covered with vegetation.




4        Conclusion

    In this paper, the effect of variations in the emissivity of underlying surface on the flux of longwave radiation
leaving the atmosphere was analysed. The approach implemented in this paper is based on the functional theory of
sensitivity. Using the analytical result obtained in the work on the coefficient of differential sensitivity of the outgoing
radiation as well as the computational complex created on the basis of the LBLRTM model, it is shown that the
maximum sensitivity of the outgoing longwave radiation to the variations of emissivity is observed in the ranges of
780–1000 cm-1 and 1015–1200 cm-1.



References
[1] Trenberth K.E., Fasullo J.T., Kiehl J. Earth’s global energy budget // Bull. Am. Meteorol. Soc. 2009. Vol. 90.
    Pp. 311–323.
[2] Stephens G.L., Li J., Wild M. et al. An update on Earth’s energy balance in light of the latest global
    observations // Nat. Geosci. 2012. Vol. 5. Pp. 691–696.
[3] Timofeev Yu.M., Vasilyev A.V. Theoretical basis of atmospheric optics. SPb: Nauka, 2003 (in Russian).
[4] Mehta A., Susskind J. Outgoing longwave radiation from the TOVS Pathfinder Path A data set // J. Geophys.
    Res. 1999. Vol. 104. N.D1, 2.193–12.212.
[5] Susskind J., Blaisdell J.M., and Iredell L. Improved methodology for surface and atmospheric soundings, error
    estimates, and quality control procedures: the atmospheric infrared sounder science team version-6 retrieval
    algorithm // J. Applied Remote Sensing. 2014. Vol. 8. Pp. 1–33.
[6] Sarmisokov Z.T., Lagutin A.A., Mordvin E.Yu. Sensitivity of the satellite thermal infrared hyperspetrometer to
    variations of atmospheric characteristic // Proс. SPIE. 2017. Vol. 10466, 104661Z.
[7] Lagutin A. A., Uchaikin V. V. The method of conjugate equations in the theory of high-energy cosmic ray
    transport. Barnaul: ASU publishing house, 2013 (in Russian).
[8] Clough S.A., Shephard M.W., Mlawer E.J. et al. Atmospheric radiative transfer modeling: a summary of the
    AER codes // JQSRT. 2005. Vol. 91. P. 233–244.