Zurich Open Repository and Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch Year: 2009 Application of the radiosity approach to the radiation balance in complex terrain Helbig, Nora Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-30798 Dissertation Published Version Originally published at: Helbig, Nora. Application of the radiosity approach to the radiation balance in complex terrain. 2009, University of Zurich, Faculty of Science. Application of the Radiosity Approach to the Radiation Balance in Complex Terrain DISSERTATION zur Erlangung der naturwissenschaftlichen Doktorw¨urde (Dr. sc. nat.) vorgelegt der Mathematisch-naturwissenschaftlichen Fakult¨at der Universit¨at Z¨urich von Nora Helbig aus Deutschland Promotionskomitee Prof. Dr. Wilfried Haeberli (Vorsitz) Dr. Michael Lehning (Leitung der Dissertation) Dr. Henning L¨owe (Leitung der Dissertation) Z¨urich 2009 Contents Summary 1 Zusammenfassung 5 I Introduction 9 1 Research introduction 11 1.1 Motivation ..................................... 11 1.2 Objectives...................................... 12 1.3 Thesisstructure................................. 13 II Theory and model definitions 15 2 Terrain discretisation and terrain parameters 17 2.1 Terrain parameters from digital height models (DHM) . .......... 17 2.2 Mutual visibility / Sun-or-shadow detection . .......... 19 3 3D-model for incident broadband solar SW and LW radiation 25 3.1 SWsurfaceradiationbalancemodel . 26 3.1.1 Astronomical relationships . 27 3.1.2 Direct and diffuse sky radiation under clear skies . 32 3.1.3 Direct and diffuse sky radiation under cloudy skies . 37 3.2 LWsurfaceradiationbalancemodel . 41 3.2.1 LWskyradiation.............................. 41 3.2.2 Parameterisation of LW terrain emissions . 42 4 3D-model for broadband SW terrain reflections and LW emissions 45 4.1 Radiosityapproach............................... 45 4.1.1 Introduction ................................ 45 4.1.2 View factor computation . 47 4.1.3 Jacobi/Gauss-Seidel iteration . 52 4.1.4 Progressive Refinement iteration . 56 4.1.5 Stoppingcriteria .............................. 57 4.2 Radiosity approach for SW radiation balance in complex terrain ....... 58 4.2.1 Progressive Refinement iteration . 63 4.2.2 Stoppingcriteria .............................. 67 4.3 Radiosity approach for LW radiation balance in complex terrain ....... 74 5 Coupled model for alpine surface processes: Alpine3D 79 5.1 Modelstructure.................................. 79 5.2 Inputdata...................................... 79 5.3 Three-dimensional radiation balance module . .......... 81 5.4 One-dimensional snow pack module: SNOWPACK . 82 5.5 Additionalmodules............................... 83 III Measurements and simulations 85 6 Mutual visibility / Sun-or-shadow detection 89 6.1 Comparison of measured and modelled horizon angles . ......... 89 7 3D-model for incident broadband solar SW and LW radiation 95 7.1 SWsurfaceradiationbalancemodel . 95 7.2 LWsurfaceradiationbalancemodel . 101 8 3D-model for broadband SW terrain reflections 105 8.1 Measurement of SW terrain reflected radiation . ........ 105 8.2 Comparison of measurements with radiosity approach model ......... 110 8.3 Comparison of measurements with isotropic view factor approach model . 113 8.4 Comparison of radiosity and isotropic view factor approachmodel . 114 8.5 Comparison of radiosity approach model and Monte Carlo model MYSTIC . 117 9 Sensitivity studies on a real topography 127 9.1 Influenceofgridcellresolutions . 127 9.2 Influence of terrain parameter extraction methods . .......... 129 9.3 Influence of view factor computation methods . ........ 132 9.4 Influence of sky view factor computation methods . ........ 134 9.5 Influence LW terrain parameterisation . 139 10 Generic behaviour of SW terrain reflected radiation in complex terrain 143 10.1 Gaussianrandomfields. 143 10.2 Influence of mean slopes and solar elevation . ......... 145 10.3 Influence of grid cell resolutions . ........ 148 10.4 Influenceofsystemsize . 151 10.5 Isotropic view factor approach . 153 10.6 Effective albedo values of complex terrain . ......... 156 IV Alpine3D application 161 11 Simulating surface temperatures of winter roads with Alpine3D 163 11.1 Inputdataandmodelsetup . 164 11.2 Roadmaintenancelogs. 168 11.3 Spatial radiation patterns in the Gotthard model domain ........... 169 11.4 Measured surface temperatures versus modelled surface temperatures . 173 V Conclusions and Outlook 179 12 Research achievements and future work 181 12.1 Achievements ................................... 181 12.2Futurework..................................... 185 List of Symbols 187 List of Figures 191 List of Tables 197 References 199 Curriculum Vitae 211 Acknowledgements 213 Summary This thesis addresses the detailed description of the radiation balance in complex terrain to improve the determination of local surface characteristics. The radiation balance in complex terrain is best described by a three-dimensional radiation balance model, which accounts for terrain influences. However, here, the radiation balance model has to remain a module of the modular model system Alpine3D for alpine surface processes. This means, the atmosphere is included by only one horizontal layer in the radiation balance model such that it can be driven by a single, exposed radiation measurement. Therefore, the development of the radi- ation balance model is restricted in the sense of complexity. In particular, the research of this thesis focussed on the development of a radiation balance model which accounts for a detailed treatment of (i) shading and (ii) multiple terrain reflec- tions and emissions but which, in the same time, remained a module of the surface process model Alpine3D. The radiation balance model in Alpine3D computes spatial incident direct and diffuse sky radiation values under cloudy sky conditions from empirical parameterisations. Those pa- rameterisations are critically reviewed, extended and partly replaced. In order to enable an accurate representation of shading from mountains as well as of visibility between surfaces more robust algorithms are implemented that determine mutual visibility between individual grid surfaces. Additionally, the representation of steep terrain ridges by digital height models is improved by applying the extraction method of terrain parameters of Corripio (2002). The radiosity approach is selected to satisfy the restrictions of the development of a detailed radiation balance model that accounts for anisotropic terrain effects but remains a module of the surface process model Alpine3D. Thereby, the radiosity approach is applied for the first time to compute the three-dimensional radiation balance in complex terrain. Its application can be seen to fill the presently large gap in complexity between three-dimensional radiation models for a cloudy atmosphere and models applying the so-called isotropic view factor ap- proach (that approximates the visible terrain by means of an isotropic surface). In contrast to the isotropic view factor approach the radiosity approach uses anisotropic view factors and does include (anisotropic) multiple terrain reflections. The radiosity equation was first intro- duced in computer graphics applying techniques originally developed in thermal engineering to describe the exchange of radiant energy between surfaces. Some numerical methods are extended and tailored to special characteristics of the radiation problem: Progressive Refinement iteration is chosen to solve the discretised radiosity equa- tion in complex terrain. The iteration order criterion of Cohen et al. (1988) with the largest unshot radiant power in the model domain is adapted to include the terrain view factor sum. A new stopping criterion was developed for the iterative solution of the linear system of the radiosity equation. This criterion can be related to the true error via some bound. 2 CONTENTS The newly developed radiation balance model is verified with point measurements and sen- sitivity studies. On the one hand, permanent radiation measurements at Weissfluhjoch study site at 2540 m a.s.l. close to Davos, Switzerland are used. For clear sky days the modelled values with the reviewed and extended clear sky parameterisations resemble the measured components well. It turned out as a main result that the lack of an adequate decomposition model for the extraction of cloudy sky radiation components out of one measured global radiation value is leading to inaccurately modelled radiation components. The measured diffuse sky radiation is mostly overestimated whereas the measured direct radiation is mostly underestimated by the modelled values. On the other hand, a new measurement is designed to characterise the terrain radiation. Modelled and measured terrain reflected radiation values agree quite well. A measured hori- zon line with a tachymeter confirms that the new visibility algorithm works well. In order to assess the radiosity approach in the context of existing model approaches a com- parison is made with both, simpler models and more sophisticated models. When compared to a full three-dimensional radiative transfer Monte Carlo model for a cloudy atmosphere, namely MYSTIC, the agreement is quite good on a clear sky day, i.e. when the clear sky parameterisations are used in the radiation balance model. When compared to simulations with the simpler isotropic view factor approach it could be shown that the radiosity approach