The Conversion from Effective Radiances to Equivalent Brightness Temperatures

The Conversion from Effective Radiances to Equivalent Brightness Temperatures Doc.No. : EUM/MET/TEN/11/0569 EUMETSAT Issue : v1 Eumetsat-Allee 1, D-64295 Darmstadt, Germany Date : 5 October 2012 Tel: +49 6151 807-7 WBS : Fax: +49 6151 807 555 http://www.eumetsat.int c@eumetsat The copyright of this document is the property of EUMETSAT

Document Signature Table Prepared by Reviewed by Reviewed by Reviewed by Name Function Signature Date Stephen Tjemkes Meteorological Scientist Rolf Stuhlmann MTG Programme Scientist Tim Hewison Johannes Müller Meteorological Scientist Instrument Data Processing Engineer Approved Volker Gärtner H/USD Release authorised by Sergio Rota H/GEO Distribution List Name No. of Copies WEB DISTRIBUTION 1

Document Change Record Issue / Date DCN. Changed Pages / Revision No. Paragraphs v1 5 October 2012 New

Contents 1 Introduction 2 Simple relation 3 Coefficients for Meteosat 8 (MSG-FM-1) and 9 (MSG-FM-2) 4 Coefficients for MSG-FM-3 and -4. 5 Summary 6 Reference Documents 1 2 3 4 5 6 7 Tables 7 7.1 General...................................... 7 7.2 Parameter tables.................................. 8 7.2.1 Table for Meteosat-8 (MSG FM-1).................... 8 7.2.2 Table for Meteosat-9 (MSG FM-2)................... 9 7.2.3 Table for MSG FM-3........................... 9 7.2.4 Table for MSG FM-4........................... 10 8 Figures 11

List of Tables evaluated using the kapteyn non-linear regression module (http://www.astro.rug.nl/software/kapteyn/inde evaluated using the kapteyn non-linear regression module ( (http://www.astro.rug.nl/software/kapteyn/ind 7.1 Values for constants at time of writing [RD.3]................... 7 7.2 Values for the regression parameters ν c, α and β of the three parameter model evaluated using the non-linear regression analysis. The input LUT of radiances and brightness temperatures was calculated for brightness temperatures between and K with steps of 0.1 K. The values are valid for SEVIRI on Meteosat 8 (MSG Flight Model 1)......................... 8 7.3 Values for the regression parameters ν c, α and β of the three parameter model evaluated using the non-linear regression analysis. The input LUT of radiances and brightness temperatures was calculated for brightness temperatures between and K with steps of 0.1 K. The values are valid for SEVIRI on Meteosat 9 (MSG Flight Model 2)......................... 9 7.4 Values for the regression parameters ν c, α and β of the three parameter model for the different channels on MSG flight model 3. The input LUT of radiances and brightness temperatures were taken from the IMPF radiance and brightness temperatures.................................... 9 7.5 Values for the regression parameters ν c, α and β of the three parameter model for the different channels on MSG flight model 4. The input LUT of radiances and brightness temperatures were taken from the IMPF radiance and brightness temperatures.................................... 10

List of Figures 8.1 Result of the non-linear regression for ir 39 on Meteosat 8 (MSG-FM-1).... 11 8.2 Result of the non-linear regression for wv 62 on Meteosat 8 (MSG-FM-1).... 12 8.3 Result of the non-linear regression for wv 73 on Meteosat 8 (MSG-FM-1).... 13 8.4 Result of the non-linear regression for ir 87 on Meteosat 8 (MSG-FM-1).... 14 8.5 Result of the non-linear regression for ir 97 on Meteosat 8 (MSG-FM-1).... 15 8.6 Result of the non-linear regression for ir 108 on Meteosat 8 (MSG-FM-1).... 16 8.7 Result of the non-linear regression for ir 120 on Meteosat 8 (MSG-FM-1).... 17 8.8 Result of the non-linear regression for ir 134 on Meteosat 8 (MSG-FM-1).... 18 8.9 Result of the non-linear regression for ir 39 on Meteosat 9 (MSG-FM-2).... 19 8.10 Result of the non-linear regression for wv 62 on Meteosat 9 (MSG-FM-2).... 20 8.11 Result of the non-linear regression for wv 73 on Meteosat 9 (MSG-FM-2).... 21 8.12 Result of the non-linear regression for ir 87 on Meteosat 9 (MSG-FM-2).... 22 8.13 Result of the non-linear regression for ir 97 on Meteosat 9 (MSG-FM-2).... 23 8.14 Result of the non-linear regression for ir 108 on Meteosat 9 (MSG-FM-2).... 24 8.15 Result of the non-linear regression for ir 120 on Meteosat 9 (MSG-FM-2).... 25 8.16 Result of the non-linear regression for ir 134 on Meteosat 9 (MSG-FM-2).... 26 8.17 Result of the non-linear regression for ir 39 on MSG FM3............ 27 8.18 Result of the non-linear regression for wv 62 on MSG FM3........... 28 8.19 Result of the non-linear regression for wv 73 on MSG FM3........... 29 8.20 Result of the non-linear regression for ir 87 on MSG FM3............ 30 8.21 Result of the non-linear regression for ir 97 on MSG FM3............ 31 8.22 Result of the non-linear regression for ir 108 on MSG FM3........... 32 8.23 Result of the non-linear regression for ir 120 on MSG FM3........... 33 8.24 Result of the non-linear regression for ir 134 on MSG FM3........... 34 8.25 Result of the non-linear regression for ir 39 on MSG FM4............ 35 8.26 Result of the non-linear regression for wv 62 on MSG FM4........... 36 8.27 Result of the non-linear regression for wv 73 on MSG FM4........... 37 8.28 Result of the non-linear regression for ir 87 on MSG FM4............ 38 8.29 Result of the non-linear regression for ir 97 on MSG FM4............ 39 8.30 Result of the non-linear regression for ir 108 on MSG FM4........... 40 8.31 Result of the non-linear regression for ir 120 on MSG FM4........... 41 8.32 Result of the non-linear regression for ir 134 on MSG FM4........... 42

Abstract This paper documents the relation between effective radiance and equivalent brightness temperature for the MSG-MPEF. It lists the rationale, method and results for a simple analytical expression to convert the observed effective radiances in mw m 2 sr 1 (cm 1 ) 1 to equivalent brightness temperatures in units kelvin.

Chapter 1 Introduction A satellite instrument measures the upwelling radiation at the top of the atmosphere. Let L ν (p t, µ) represents the spectral variation of the upwelling radiance at the top of the atmosphere (at pressure p t ) propagating towards the satellite sensor, r ν represent the spectral response function, ν wavenumber and µ = cos θ, with θ is the angle between zenith and viewing direction, then the observed effective radiance L is given by: L = r ν L ν (p t, µ)dν/ r ν dν. (1.1) Δν These observations are expressed in mw m 2 sr 1 (cm 1 ) 1. To facilitate an easy interpre- tation of these observations, these units are converted into equivalent black body temperatures (in kelvin-units) or alternatively known as equivalent brightness temperatures. The brightness temperature is defined as the temperature of a black body which emits the same amount of radiation as observed. Thus the brightness temperature (T b ) follows from L = rν B (ν, T b ) dν/ r ν dν. (1.2) Δν with B the Planck-function. Inversion of Eq. 1.2 is not possible because of the variation of the Planck and the spectral response function (r ν ) with wavenumber. It is possible to generate a pre-calculated lookup table (LUT) which contains the calculated effective radiance of a black body at a given temperature as observed by the particular instrument using Eq. 1.2. The brightness temperature of an actual observation is found from interpolation in this LUT. This is the method currently adopted by the IMage Processing Facility (IMPF) within the Meteosat Second Generation Ground Segment (MSG-GS) as documented in [RD.1].

Chapter 2 Simple relation For several applications the search in a pre-calculated LUT is too time consuming and for these a simple relation between observed effective radiance and the corresponding brightness temperature is preferred over the interpolation in a pre-calculated LUT. A simple relation can be derived assuming that the Planck-function is independent of wavenumber over the particular interval: L = r ν B (ν, T b ) dν/ r ν dν. Δν (2.1) B (T b ) where B represents a mean value over the interval. Upon substitution in the Planck-function, the simple relation reads: L 2hc 2 ν 3 c / (exp (hcν c /kt b ) 1), (2.2) where ν c represents a representative wavenumber, h the Planck constant, c speed of light and k the Boltzmann constant. Equation 2.2 is a single parameter function. The single parameter (ν c ) can be found from a non-linear regression of the pre-calculated LUT mentioned above. An alternative relation can be found from a generalization of Eq. 2.2: L γ/ (exp (δ/t b ) 1), (2.3) with γ and δ two parameters which can be found from a non-linear regression of the precalculated LUT mentioned above. More often a three parameter function is fitted to the LUT: L 2hc 2 ν 3 c / (exp (hcν c /k [αt b + β]) 1), (2.4) with ν c, α and β three parameters which can be found from a non-linear regression of the precalculated LUT mentioned above. This three parameter function is adopted for the MSG-MPEF, because it has been shown to be superior over the performance of the other two models.

Chapter 3 Coefficients for Meteosat 8 (MSG-FM-1) and 9 (MSG-FM-2) LUT s were generated for all the eight infrared channels of the SEVIRI instruments on Meteosat 8 (Meteosat Second Generation Flight Model 1, MSG-FM-1) and Meteosat 9 (MSG-FM-2) for the brightness temperature domain between and K with increments of 0.1 K. These LUT s were then used to derive the parameters of the non-linear regression model (eq. 2.4). The regression coefficients are listed in tables 7.2 and 7.3 for the Meteosat-8 and Meteosat-9 respectively. The goodness of the derivation of the parameters are presented as a series of figures for each of the channels on both Meteosat-8 and -9. Figures 8.1-8.8 show a series of three panel figures. The top panel shows the variation of the brightness temperature as a function of effective radiances as provided by the non-linear regression model, with the parameters settings as presented in this panel. Also shown in this figure are selected entries from the pre-calculated look-up table indicated by the red symbols. The lower two panels represent the residuals between the LUT and the EBBT calculated with the three parametric non-linear regression model (2.4), with the parameters in tables 7.2 and 7.3 for the SEVIRI channels on Meteosat-8 (MSG-FM-1) and Meteosat-9 (MSG-FM-2) respectively. The bottom left panel shows the effective radiance residuals as function of effective radiances, while the right bottom panel shows the residuals as function of equivalent brightness temperatures. These panel shows that the brightness temperature residuals are a smooth function of radiance, and a root mean square difference of the order of 0.01 K or less is found, indicating the goodness of the approximation.

Chapter 4 Coefficients for MSG-FM-3 and -4. Since the beginning of this project there has been several changes in relation to the conversion of effective radiances to brightness temperature. In 2007/2008 EUMETSAT ran a study with the SEVIRI manufacturer to reprocess all SE VIRI ground characterization data in order to obtain a fully consistent set of instrument characterization in terms of effective radiance. One justification was to remove existing inconsistencies in the communication to the users. The study results were compiled into an IMPF software and database re-design. Among other things this compilation contains spacecraft specific tables to relate temperatures to radiances as documented [RD.1] Since 2008, EUMETSAT answering to user enquiries in that respect has been with reference to the published tables. In line with this, the three parameter for the non-linear regression function for MSG-FM-3 and FM-4 have been generated to reproduce this table as accurately as possible. The second important modification is that an alternative method to derive the non-linear regression parameters was implemented. The availability of the scripting language python allowed for a portable implementation of a non-linear regression method relatively simply. Although the intrinsic non-linear fitting routines will allow an accurate determination of the three parameters, the routines adopted here are taken from a collection of Python modules and applications developed by the computer group of the Kapteyn Astronomical Institute, University of Groningen, The Netherlands ([RD.2]) as this package provided some additional diagnostic tools. The results for MSG-FM-3 and MSG-FM-4 are listed in tables 7.4 and 7.5. The goodness of the fits is shown in a similar way as discussed in section 3.

Chapter 5 Summary The equivalent brightness temperature of an satellite observation, is defined as the temperature of a black body which emits the same amount of radiation as observed. Thus the brightness temperature follows from, L = r ν L ν (p t, µ)dν/ r ν dν. (5.1) Δν with L the observed radiances (in mw m 2 sr 1 (cm 1 ) 1 ), B the Planck function, T b the equivalent brightness temperature (in kelvin units), ν wavenumber (in cm-1), and r ν the instrument spectral response. In the MSG-MPEF the following simple relation between the SEVIRI radiances and the equivalent brightness temperature is adopted: L C 1 ν 3 c / (exp (C 2 ν c / [αt b + β]) 1), (5.2) with C 1 = 2hc 2 and C 2 = hc/k radiation constants where c, h, and k are the speed of light, Planck, and Boltzmann constant respectively (Values for these constants are listed in Table 7.1). The inverse of equation 5.2 reads C 2 ν c β T b =, (5.3) α log C 1 ν 3 c /R + 1 α The regression coefficients ν c, α and β are found from a non-linear regression of the precalculated lookup table, generated using equation 5.1, for the different SEVIRI channels. For Meteosat-8 and -9 this look-up table was generated as part of this process to determine the three parameters, wereas for MSG-FM-3 and FM-4 the lookup tables prepared by the IMPF were adopted. Results of the non-linear regression are shown in tables 7.2-7.3 for Meteosat-8 and -9 respectively and in tables 7.4-7.5 for MSG-Flight model 3 and 4 respectively. The goodness of the fit to the lookup tables is shown in figures 8.17 8.24 and figures 8.25 8.32 for MSG-FM3 and MSG-FM4 respectively.

Chapter 6 Reference Documents [RD.1] Effective Radiances and Brightness Temperature Relation Tables for Meteosat Second Generation. EUM/OPS-MSG/TEN/08/0024, Issue v2. Available through http://www.eumetsat.int. [RD.2] http://www.astro.rug.nl/software/kapteyn/index.html. [RD.3] The 2010 CODATA from the NIST reference on Constants, Units and Uncertainty, from NIST Physical Measurement Laboratory http://www.nist.gov/cuu/index.html.

Chapter 7 Tables 7.1 General Table 7.1: Values for constants at time of writing [RD.3]. Constant Speed of light in vacuum Planck Bolzmann Value 299792458 ms 1 6.62606957 10 34 Js 1.3806488 10 23 JK 1

7.2 Parameter tables 7.2.1 Table for Meteosat-8 (MSG FM-1) Table 7.2: Values for the regression parameters ν c, α and β of the three parameter model evaluated using the nonlinear regression analysis. The input LUT of radiances and brightness temperatures was calculated for brightness temperatures between and K with steps of 0.1 K. The values are valid for SEVIRI on Meteosat 8 (MSG Flight Model 1). Channel No. Channel ID a ν c α β b 4 5 6 7 8 9 10 11 IR 3.9 WV 6.2 WV 7.3 IR 8.7 IR 9.7 IR 10.8 IR 12.0 IR 13.4 2567.330 1598.103 1362.081 1149.069 1034.343 930.647 839.660 752.387 0.9956 0.9962 0.9991 0.9996 0.9999 0.9983 0.9988 0.9981 3.410 2.218 0.478 0.179 0.060 0.625 0.397 0.578 a Units: cm 1. b Units: K.

7.2.2 Table for Meteosat-9 (MSG FM-2) Table 7.3: Values for the regression parameters ν c, α and β of the three parameter model evaluated using the nonlinear regression analysis. The input LUT of radiances and brightness temperatures was calculated for brightness temperatures between and K with steps of 0.1 K. The values are valid for SEVIRI on Meteosat 9 (MSG Flight Model 2). Channel No. Channel ID a ν c α β b 4 5 6 7 8 9 10 11 IR 3.9 WV 6.2 WV 7.3 IR 8.7 IR 9.7 IR 10.8 IR 12.0 IR 13.4 2568.832 1600.548 1360.330 1148.620 1035.289 931.700 836.445 751.792 0.9954 0.9963 0.9991 0.9996 0.9999 0.9983 0.9988 0.9981 3.438 2.185 0.470 0.179 0.056 0.640 0.408 0.561 a Units: cm 1. b Units: K. 7.2.3 Table for MSG FM-3 Table 7.4: Values for the regression parameters ν c, α and β of the three parameter model evaluated using the kapteyn non-linear regression module (http://www.astro.rug.nl/software/kapteyn/index.html) for the different channels on MSG flight model 3. The input LUT of radiances and brightness temperatures were taken from the IMPF radiance and brightness temperatures. Channel No. Channel ID a ν c α β b 4 5 6 7 8 9 10 11 IR 3.9 WV 6.2 WV 7.3 IR 8.7 IR 9.7 IR 10.8 IR 12.0 IR 13.4 2547.771 1595.621 1360.377 1148.130 1034.715 929.842 838.659 750.653 0.9915 0.9960 0.9991 0.9996 0.9999 0.9983 0.9988 0.9982 2.9002 2.0337 0.4340 0.1714 0.0527 0.6084 0.3882 0.5390 a Units: cm 1. b Units: K.

7.2.4 Table for MSG FM-4 Table 7.5: Values for the regression parameters ν c, α and β of the three parameter model evaluated using the kapteyn non-linear regression module ( (http://www.astro.rug.nl/software/kapteyn/index.html)) for the different channels on MSG flight model 4. The input LUT of radiances and brightness temperatures were taken from the IMPF radiance and brightness temperatures. Channel No. Channel ID a ν c α β b 4 5 6 7 8 9 10 11 IR 3.9 WV 6.2 WV 7.3 IR 8.7 IR 9.7 IR 10.8 IR 12.0 IR 13.4 2555.280 1596.080 1361.748 1147.433 1034.851 931.122 839.113 748.585 0.9916 0.9959 0.9990 0.9996 0.9998 0.9983 0.9988 0.9981 2.9438 2.0780 0.4929 0.1731 0.0597 0.6256 0.4002 0.5635 a Units: cm 1. b Units: K.

Chapter 8 Figures Fitted Curve channel ir_39msg1 200 Alternative: a=0.9956, b=3.4, v c =2567.3 R =c 1 v 3 c /(exp((c 2 ν c /(a T b +b)) 1) 0 1 2 3 4 5 6 0.01 Difference between LUT and fitted curve 0.01 Difference between LUT and fitted curve msg1 ir_39 0.00 0.00 0.01 0.01 0.02 0.03 0.02 0.03 0.04 0.04 0.05 0.05 0.06 0 1 2 3 4 5 6 0.06 200 Figure 8.1: Result of the non-linear regression for ir 39 on Meteosat 8 (MSG-FM-1).

Fitted Curve channel wv_62msg1 200 Alternative: a=0.9962, b=2.2180, v c =1598.1030 R =c 1 v 3 c /(exp((c 2 ν c /(a T b +b)) 1) 0 10 20 30 40 50 60 70 0.005 Difference between LUT and fitted curve 0.005 Difference between LUT and fitted curve msg1 wv_62 0.010 0.010 0.015 0.015 0.020 0.025 0.030 0.020 0.025 0.030 0.035 0.035 0.040 0.040 0.045 0 10 20 30 40 50 60 70 0.045 200 Figure 8.2: Result of the non-linear regression for wv 62 on Meteosat 8 (MSG-FM-1).

Fitted Curve channel wv_73msg1 Alternative: a=0.9991, b=0.4780, v c =1362.0810 R =c 1 vc 3 /(exp((c 2 ν c /(a T b +b)) 1) 200 0 20 40 60 80 120 Difference between LUT and fitted curve Difference between LUT and fitted curve msg1 wv_73 0.0015 0.0015 0.0020 0.0020 0.0025 0 20 40 60 80 120 0.0025 200 Figure 8.3: Result of the non-linear regression for wv 73 on Meteosat 8 (MSG-FM-1).

Fitted Curve channel ir_87msg1 Alternative: a=0.9996, b=0.1790, v c =1149.0690 R =c 1 vc 3 /(exp((c 2 ν c /(a T b +b)) 1) 200 0 20 40 60 80 120 140 160 180 0.0014 Difference between LUT and fitted curve 0.0014 Difference between LUT and fitted curve msg1 ir_87 0.0012 0.0012 0.0008 0.0006 0.0008 0.0006 0.0004 0.0004 0.0002 0.0002 0 20 40 60 80 120 140 160 180 200 Figure 8.4: Result of the non-linear regression for ir 87 on Meteosat 8 (MSG-FM-1).

Fitted Curve channel ir_97msg1 Alternative: a=0.9999, b=0.0600, v c =1034.3430 R =c 1 vc 3 /(exp((c 2 ν c /(a T b +b)) 1) 200 0 50 200 0.006 Difference between LUT and fitted curve 0.006 Difference between LUT and fitted curve msg1 ir_97 0.008 0.008 0.010 0.012 0.010 0.012 0.014 0.014 0.016 0 50 200 0.016 200 Figure 8.5: Result of the non-linear regression for ir 97 on Meteosat 8 (MSG-FM-1).

Fitted Curve channel ir_108msg1 Alternative: a=0.9983, b=0.6, v c =930.6470 R =c 1 vc 3 /(exp((c 2 ν c /(a T b +b)) 1) 200 0 50 200 0.007 Difference between LUT and fitted curve 0.007 Difference between LUT and fitted curve msg1 ir_108 0.006 0.006 0.005 0.005 0.004 0.003 0.004 0.003 0.002 0.002 0.001 0 50 200 0.001 200 Figure 8.6: Result of the non-linear regression for ir 108 on Meteosat 8 (MSG-FM-1).

Fitted Curve channel ir_120msg1 Alternative: a=0.9988, b=0.3970, v c =839.6600 R =c 1 vc 3 /(exp((c 2 ν c /(a T b +b)) 1) 200 0 50 200 0.006 Difference between LUT and fitted curve 0.006 Difference between LUT and fitted curve msg1 ir_120 0.005 0.005 0.004 0.004 0.003 0.002 0.003 0.002 0.001 0.001 0.000 0.000 0.001 0 50 200 0.001 200 Figure 8.7: Result of the non-linear regression for ir 120 on Meteosat 8 (MSG-FM-1).

Fitted Curve channel ir_134msg1 Alternative: a=0.9981, b=0.5780, v c =752.3870 R =c 1 vc 3 /(exp((c 2 ν c /(a T b +b)) 1) 200 0 50 200 0.0035 Difference between LUT and fitted curve 0.0035 Difference between LUT and fitted curve msg1 ir_134 0.0040 0.0040 0.0045 0.0050 0.0045 0.0050 0.0055 0.0055 0.0060 0 50 200 0.0060 200 Figure 8.8: Result of the non-linear regression for ir 134 on Meteosat 8 (MSG-FM-1).

Fitted Curve channel ir_39msg2 200 Alternative: a=0.9954, b=3.4380, v c =2568.8320 R =c 1 v 3 c /(exp((c 2 ν c /(a T b +b)) 1) 0 1 2 3 4 5 6 0.01 Difference between LUT and fitted curve 0.01 Difference between LUT and fitted curve msg2 ir_39 0.00 0.00 0.01 0.01 0.02 0.03 0.04 0.02 0.03 0.04 0.05 0.05 0.06 0.06 0.07 0 1 2 3 4 5 6 0.07 200 Figure 8.9: Result of the non-linear regression for ir 39 on Meteosat 9 (MSG-FM-2).

Fitted Curve channel wv_62msg2 200 Alternative: a=0.9963, b=2.1850, v c =1600.5480 R =c 1 v 3 c /(exp((c 2 ν c /(a T b +b)) 1) 0 10 20 30 40 50 60 70 0.000 Difference between LUT and fitted curve 0.000 Difference between LUT and fitted curve msg2 wv_62 0.002 0.002 0.004 0.004 0.006 0.006 0.008 0.010 0.012 0.008 0.010 0.012 0.014 0.014 0.016 0.016 0.018 0 10 20 30 40 50 60 70 0.018 200 Figure 8.10: Result of the non-linear regression for wv 62 on Meteosat 9 (MSG-FM-2).

Fitted Curve channel wv_73msg2 Alternative: a=0.9991, b=0.4700, v c =1360.3 R =c 1 vc 3 /(exp((c 2 ν c /(a T b +b)) 1) 200 0 20 40 60 80 120 0.0030 Difference between LUT and fitted curve 0.0030 Difference between LUT and fitted curve msg2 wv_73 0.0025 0.0025 0.0020 0.0020 0.0015 0.0015 0 20 40 60 80 120 200 Figure 8.11: Result of the non-linear regression for wv 73 on Meteosat 9 (MSG-FM-2).

Fitted Curve channel ir_87msg2 Alternative: a=0.9996, b=0.1790, v c =1148.6200 R =c 1 vc 3 /(exp((c 2 ν c /(a T b +b)) 1) 200 0 20 40 60 80 120 140 160 180 0.0012 Difference between LUT and fitted curve 0.0012 Difference between LUT and fitted curve msg2 ir_87 0.0008 0.0008 0.0006 0.0004 0.0002 0.0006 0.0004 0.0002 0.0002 0.0002 0.0004 0 20 40 60 80 120 140 160 180 0.0004 200 Figure 8.12: Result of the non-linear regression for ir 87 on Meteosat 9 (MSG-FM-2).

Fitted Curve channel ir_97msg2 Alternative: a=0.9999, b=0.0560, v c =1035.2890 R =c 1 vc 3 /(exp((c 2 ν c /(a T b +b)) 1) 200 0 50 200 0.005 Difference between LUT and fitted curve 0.005 Difference between LUT and fitted curve msg2 ir_97 0.006 0.006 0.007 0.007 0.008 0.009 0.010 0.008 0.009 0.010 0.011 0.011 0.012 0.012 0.013 0 50 200 0.013 200 Figure 8.13: Result of the non-linear regression for ir 97 on Meteosat 9 (MSG-FM-2).

Fitted Curve channel ir_108msg2 Alternative: a=0.9983, b=0.6400, v c =931.7000 R =c 1 vc 3 /(exp((c 2 ν c /(a T b +b)) 1) 200 0 50 200 0.0030 Difference between LUT and fitted curve 0.0030 Difference between LUT and fitted curve msg2 ir_108 0.0035 0.0035 0.0040 0.0040 0.0045 0.0050 0.0045 0.0050 0.0055 0.0055 0.0060 0.0060 0.0065 0 50 200 0.0065 200 Figure 8.14: Result of the non-linear regression for ir 108 on Meteosat 9 (MSG-FM-2).

Fitted Curve channel ir_120msg2 Alternative: a=0.9988, b=0.4080, v c =836.4450 R =c 1 vc 3 /(exp((c 2 ν c /(a T b +b)) 1) 200 0 50 200 0.0076 Difference between LUT and fitted curve 0.0076 Difference between LUT and fitted curve msg2 ir_120 0.0078 0.0078 0.0080 0.0080 0.0082 0.0084 0.0082 0.0084 0.0086 0.0086 0.0088 0 50 200 0.0088 200 Figure 8.15: Result of the non-linear regression for ir 120 on Meteosat 9 (MSG-FM-2).

Fitted Curve channel ir_134msg2 Alternative: a=0.9981, b=0.5610, v c =751.7920 R =c 1 vc 3 /(exp((c 2 ν c /(a T b +b)) 1) 200 0 50 200 0.014 Difference between LUT and fitted curve 0.014 Difference between LUT and fitted curve msg2 ir_134 0.012 0.012 0.010 0.010 0.008 0.006 0.008 0.006 0.004 0.004 0.002 0 50 200 0.002 200 Figure 8.16: Result of the non-linear regression for ir 134 on Meteosat 9 (MSG-FM-2).

400 Fitted Curve channel msg3 ir_39 200 IMPF-fit a=0.9915, R =c 1 vc 3 /(exp((c b=2.9002, v c =2547.7708 2 ν c /(a T b +b)) 1) 50 0 1 2 3 4 5 6 Difference between LUT and fitted curve msg3 ir_39 Difference between LUT and fitted curve msg3 ir_39 0.02 0.02 0.00 0.00 0.02 0.02 0.04 0.04 0.06 0.06 0 1 2 3 4 5 6 200 Figure 8.17: Result of the non-linear regression for ir 39 on MSG FM3.

400 Fitted Curve channel msg3 wv_62 200 IMPF-fit a=0.9960, R =c 1 vc 3 /(exp((c b=2.0337, v c =1595.6210 2 ν c /(a T b +b)) 1) 50 0 10 20 30 40 50 60 70 0.010 Difference between LUT and fitted curve msg3 wv_62 0.010 Difference between LUT and fitted curve msg3 wv_62 0.005 0.005 0.000 0.000 0.005 0.010 0.005 0.010 0.015 0.015 0.020 0.020 0.025 0 10 20 30 40 50 60 70 0.025 200 Figure 8.18: Result of the non-linear regression for wv 62 on MSG FM3.

400 Fitted Curve channel msg3 wv_73 200 IMPF-fit a=0.9991, R =c 1 vc 3 /(exp((c b=0.4340, v c =1360.3767 2 ν c /(a T b +b)) 1) 50 0 20 40 60 80 120 Difference between LUT and fitted curve msg3 wv_73 Difference between LUT and fitted curve msg3 wv_73 0.0015 0 20 40 60 80 120 0.0015 200 Figure 8.19: Result of the non-linear regression for wv 73 on MSG FM3.

400 Fitted Curve channel msg3 ir_87 IMPF-fit a=0.9996, R =c 1 vc 3 200 /(exp((c b=0.1714, v c =1148.1299 2 ν c /(a T b +b)) 1) 50 0 20 40 60 80 120 140 160 180 0.0003 Difference between LUT and fitted curve msg3 ir_87 0.0003 Difference between LUT and fitted curve msg3 ir_87 0.0002 0.0002 0.0001 0.0001 0.0001 0.0002 0.0001 0.0002 0.0003 0.0003 0.0004 0.0004 0 20 40 60 80 120 140 160 180 200 Figure 8.20: Result of the non-linear regression for ir 87 on MSG FM3.

400 Fitted Curve channel msg3 ir_97 IMPF-fit a=0.9999, R =c 1 vc 3 200 /(exp((c b=0.0527, v c =1034.7153 2 ν c /(a T b +b)) 1) 50 0 50 200 0.00010 Difference between LUT and fitted curve msg3 ir_97 0.00010 Difference between LUT and fitted curve msg3 ir_97 5 5 0 5 0 5 0.00010 0.00010 0.00015 0 50 200 0.00015 200 Figure 8.21: Result of the non-linear regression for ir 97 on MSG FM3.

400 Fitted Curve channel msg3 ir_108 IMPF-fit a=0.9983, R =c 1 vc 3 200 /(exp((c b=0.6084, v c =929.8417 2 ν c /(a T b +b)) 1) 50 0 50 200 0.0015 Difference between LUT and fitted curve msg3 ir_108 0.0015 Difference between LUT and fitted curve msg3 ir_108 0.0015 0.0015 0.0020 0.0020 0.0025 0 50 200 0.0025 200 Figure 8.22: Result of the non-linear regression for ir 108 on MSG FM3.

400 Fitted Curve channel msg3 ir_120 IMPF-fit a=0.9988, R =c 1 vc 3 200 /(exp((c b=0.3882, v c =838.6594 2 ν c /(a T b +b)) 1) 50 0 50 200 Difference between LUT and fitted curve msg3 ir_120 Difference between LUT and fitted curve msg3 ir_120 0.0015 0 50 200 0.0015 200 Figure 8.23: Result of the non-linear regression for ir 120 on MSG FM3.

400 Fitted Curve channel msg3 ir_134 IMPF-fit a=0.9982, R =c 1 vc 3 200 /(exp((c b=0.5390, v c =750.6529 2 ν c /(a T b +b)) 1) 50 0 50 200 0.0015 Difference between LUT and fitted curve msg3 ir_134 0.0015 Difference between LUT and fitted curve msg3 ir_134 0.0015 0.0015 0.0020 0.0020 0.0025 0.0025 0.0030 0 50 200 0.0030 200 Figure 8.24: Result of the non-linear regression for ir 134 on MSG FM3.

400 Fitted Curve channel msg4 ir_39 200 IMPF-fit a=0.9916, R =c 1 vc 3 /(exp((c b=2.9438, v c =2555.2800 2 ν c /(a T b +b)) 1) 50 0 1 2 3 4 5 6 Difference between LUT and fitted curve msg4 ir_39 Difference between LUT and fitted curve msg4 ir_39 0.02 0.02 0.00 0.00 0.02 0.02 0.04 0.04 0.06 0.06 0 1 2 3 4 5 6 200 Figure 8.25: Result of the non-linear regression for ir 39 on MSG FM4.

400 Fitted Curve channel msg4 wv_62 200 IMPF-fit a=0.9959, R =c 1 vc 3 /(exp((c b=2.0780, v c =1596.0799 2 ν c /(a T b +b)) 1) 50 0 10 20 30 40 50 60 70 0.010 Difference between LUT and fitted curve msg4 wv_62 0.010 Difference between LUT and fitted curve msg4 wv_62 0.005 0.005 0.000 0.000 0.005 0.010 0.005 0.010 0.015 0.015 0.020 0 10 20 30 40 50 60 70 0.020 200 Figure 8.26: Result of the non-linear regression for wv 62 on MSG FM4.

400 Fitted Curve channel msg4 wv_73 200 IMPF-fit a=0.9990, R =c 1 vc 3 /(exp((c b=0.4929, v c =1361.7482 2 ν c /(a T b +b)) 1) 50 0 20 40 60 80 120 Difference between LUT and fitted curve msg4 wv_73 Difference between LUT and fitted curve msg4 wv_73 0.0015 0.0015 0.0020 0 20 40 60 80 120 0.0020 200 Figure 8.27: Result of the non-linear regression for wv 73 on MSG FM4.

400 Fitted Curve channel msg4 ir_87 IMPF-fit a=0.9996, R =c 1 vc 3 200 /(exp((c b=0.1731, v c =1147.4332 2 ν c /(a T b +b)) 1) 50 0 20 40 60 80 120 140 160 180 0.0003 Difference between LUT and fitted curve msg4 ir_87 0.0003 Difference between LUT and fitted curve msg4 ir_87 0.0002 0.0002 0.0001 0.0001 0.0001 0.0002 0.0001 0.0002 0.0003 0.0003 0.0004 0.0004 0 20 40 60 80 120 140 160 180 200 Figure 8.28: Result of the non-linear regression for ir 87 on MSG FM4.

400 Fitted Curve channel msg4 ir_97 IMPF-fit a=0.9998, R =c 1 vc 3 200 /(exp((c b=0.0597, v c =1034.8511 2 ν c /(a T b +b)) 1) 50 0 50 200 0.00010 Difference between LUT and fitted curve msg4 ir_97 0.00010 Difference between LUT and fitted curve msg4 ir_97 5 5 0 0 5 0.00010 5 0.00010 0.00015 0.00015 0.00020 0 50 200 0.00020 200 Figure 8.29: Result of the non-linear regression for ir 97 on MSG FM4.

400 Fitted Curve channel msg4 ir_108 IMPF-fit a=0.9983, R =c 1 vc 3 200 /(exp((c b=0.6256, v c =931.1221 2 ν c /(a T b +b)) 1) 50 0 50 200 0.0015 Difference between LUT and fitted curve msg4 ir_108 0.0015 Difference between LUT and fitted curve msg4 ir_108 0.0015 0.0015 0.0020 0.0020 0.0025 0.0025 0.0030 0 50 200 0.0030 200 Figure 8.30: Result of the non-linear regression for ir 108 on MSG FM4.

400 Fitted Curve channel msg4 ir_120 IMPF-fit a=0.9988, R =c 1 vc 3 200 /(exp((c b=0.4002, v c =839.1132 2 ν c /(a T b +b)) 1) 50 0 50 200 Difference between LUT and fitted curve msg4 ir_120 Difference between LUT and fitted curve msg4 ir_120 0.0015 0 50 200 0.0015 200 Figure 8.31: Result of the non-linear regression for ir 120 on MSG FM4.

400 Fitted Curve channel msg4 ir_134 IMPF-fit a=0.9981, R =c 1 vc 3 200 /(exp((c b=0.5635, v c =748.5852 2 ν c /(a T b +b)) 1) 50 0 50 200 0.002 Difference between LUT and fitted curve msg4 ir_134 0.002 Difference between LUT and fitted curve msg4 ir_134 0.001 0.001 0.000 0.000 0.001 0.002 0.001 0.002 0.003 0.003 0.004 0 50 200 0.004 200 Figure 8.32: Result of the non-linear regression for ir 134 on MSG FM4.