the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 02 Jun 2025
A simple step heating approach for wall surface temperature estimation in the SOlar and LongWave Environmental Irradiance Geometry (SOLWEIG) model
Abstract. The urban climate is highly influenced by its building geometry, material characteristics, street orientation and high fraction of impermeable surfaces. All of these influence the microclimate and the resulting outdoor thermal comfort. Mean radiant temperature (Tmrt) is often used as an estimator for heat exposure as it is one of the most important variables governing outdoor human thermal comfort on clear, calm and warm days. The highest values of Tmrt are commonly found in front of sunlit facades where a human is exposed to high levels of direct and reflected shortwave radiation from the sun, as well as high levels of longwave radiation emitted from surrounding sunlit walls. As a consequence, outdoor thermal comfort modelling requires accurate simulation of wall surface temperatures (Ts).
The aim of this study is to present a step heating approach for calculating wall Ts in the SOlar and LongWave Environmental Irradiance Geometry model (SOLWEIG) and quantifying how it influences Tmrt. This method requires information on material characteristics, i.e. specific heat capacity, density, thermal conductivity, albedo and thickness of the outer layer of the wall, as well as radiation balance at the wall surface, and ambient air temperature. Simulated Ts is compared to observed Ts of two white walls (albedo = 0.5) in Gothenburg, Sweden; one wooden wall and one plaster brick wall. The simulations show high agreement with the 15,394 observations, with R2 = 0.93 and RMSE = 2.09 °C for the wooden wall and R2 = 0.94 and RMSE = 1.94 °C for the plaster brick wall. For the walls presented here, this new parameterization scheme results in differences in Tmrt of up to 2.5 °C compared to the previous version of SOLWEIG.
With this new approach SOLWEIG can be used to evaluate the effect of building materials on outdoor thermal comfort. The speed and accuracy of this approach suggests that it also could be applied in other areas where Ts of walls are important, for example building energy models and urban energy balance models.
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Status: final response (author comments only)
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RC1: 'Comment on egusphere-2025-2093', Anonymous Referee #1, 31 Aug 2025
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AC2: 'Reply on RC1', Nils Wallenberg, 26 Nov 2025
General Comments
In this paper, a new parameterization scheme for the SOLWEIG model is introduced and tested for a dataset from two walls with similar aspect and albedo, but different thermal properties. The evaluation shows good performance for the wall temperatures and a demonstration is made of the impact on SOLWEIG-predicted MRT. Overall, I think the work provides the basis for a useful contribution and the introduction of the scheme will broaden the capabilities of the SOLWEIG model. The step heating approach chosen seems to be based in part on calculation efficiency, although tests are not made to demonstrate its performance in this respect and I thought there was scope to include some sensitivity testing of the choice of thermal parameters. More generally, it was not clear to me if there are any assumptions implicit in the step heating approach that might impact its ability to represent environmental temperature time series. The basis of that approach seems to be in the laboratory testing of thermal properties of materials where a step change in heat input can be applied, but this may be different from the type of input from environmental radiative forcings. I recall here the comparison of PDE and Force Restore approaches for surface temperature modeling in Johnson et al. 1991 Boundary Layer Meteorology 56, 275-294 that dealt with some similar issues.
Reply:
Thank you very much for your review. It will help us clarify our manuscript. The step heating approach requires a radiation balance at the surface, i.e. incoming radiation (short and long) as well as outgoing longwave radiation based on its surface temperature. This is a simple method compared to conduction models that would require more computations. The sensitivity test will definitely be an important improvement of the manuscript, and we are working on implementing this. The basis of the approach is indeed in a laboratory setting, where a slab would be subjected to a heat pulse starting when the slab is in balance with its environment. A wall surface in an outdoor setting is, as you mention, absolutely different. In our view it would constantly be subjected to a “heat pulse” that is also changing with weather, and never in balance. The paper by Johnson et al. (1991) is a good starting point to explain this in our discussion.
Specific CommentsLine 23 Does the choice of walls with albedo = 0.5 make the test somewhat conservative for Ts, given Ts would be expected to be lower in this case (compared to a wall with lower albedo?) Perhaps this choice is related to expected MRT?
Answer:
This is possible. We are working on a sensitivity analysis that will be included in the revised version of the manuscript. This will hopefully give some answers to this. The thermal properties of the wall surface will of course influence the resulting Ts and therefore Tmrt. In our comparison with the new and old versions of SOLWEIG (or approaches for wall Ts) are based on the same albedo (0.5). What differs, then, is that the previous version of SOLWEIG uses the very simple empirical relationship by Bogren et al (2000) and Lindberg & Grimmond (2016). It is likely that difference in Tmrt would be different with another albedo, but not solely.
Line 55-57 The Fourier's law of diffusion is just for the conduction of heat - so a surface energy balance approach could potentially replace Fourier's law with an alternative -e.g. a force restore approach could be used to resolve conduction while still using a surface energy balance approach.
Answer:
Thank you for this, we will add a few sentences to this (e.g. TARGET).
Line 62 And possibly also to the season or time period of that calibration.
Answer:
Yes, indeed.
Line 67 Thermal effussivity is equivalent to thermal admittance (see e.g. Oke et al. 2017 Urban Climates) - which may be familiar to readers with an urban climate background so it could be useful to point out this equivalence.
Answer:
Thank you for this comment. Yes, this is true and confused us a bit in the beginning, as we were looking at thermal admittance in Oke et al. 2017 Urban Climates in early stages of our development of this model. This will be clarified.
Line 76-79. I wonder if a bit more context is needed - e.g. SOLWEIG is not quite a full energy balance model - and success at modeling Ts will also be related in part to the performance of the convective and conductive parameterizations (as well as radiation).
Answer:
Yes, this is true. SOLWEIG, is only taking radiation fluxes into account as you point out. The old parameterization for wall Ts was very simple, based on empirical relationships as in Bogren et al. (2000) and Lindberg et al. (2016) based on ground surfaces, hence, the over and underestimation presented in e.g. Kantor et al. 2020. Hopefully, our new scheme can improve the performance related to their observations.
Line 84 Are there any issues given that the environmental forcing is different from a Dirac heat pulse as assumed in the reference?
Answer:
There are definitely issues. One would be the timeframe, where in our case a wall would always be subjected to a “heat pulse”, i.e. its environment be it longwave and/or shortwave radiation, whereas in the reference they are in a very controlled lab environment where a slab would be subjected to a heat pulse for only a few seconds. We will clarify this in the discussion.
Equation 1: Equation 1) has surface temperature on both sides of the equation, since outgoing longwave that is part of omega is determined by Ts – this may be worth noting.
Answer:
Yes, and it is also mentioned in the end of section 2.1 and under 2.2.2 in relation to equation 14.
Line 102 The matching is probably best for walls a few hours after sunset (e.g. to allow west-facing walls that may have been directly irradiated to cool closer to Tair).
Answer:
Yes, this is true. Although not included in the manuscript, we have tried and gotten relatively good results from very few time steps after starting (even in daytime), which could be related to the fact that we run the equation twice for every time step to get a more “updated” Ts. It could also be related to the fact that heat storage is not included in the parameterization. But preferably, as we state in our manuscript, the model should start at nighttime. Maybe it would be good to clarify that it should start a few hours after sunset.
Equation 13 I guess fsun could also depend on the canyon geometry? and some of the empiricism is related to expected obstructions within a canyon and/or microscale facet structure such as awnings that creates self shading?
Answer:
Yes, this is exactly what we mean. But to maintain computational speed in the model we use this simple relationship. It is definitely subject to further improvement. At the same time we think that the end product of SOLWEIG, i.e. Tmrt, is not influenced too much by this, but rather by other factors. We will clarify further.
Equation 15 I'm not sure why the "fraction" terminology is used when the other values are view factors? Essentially you are calculating the view factor of the opposite wall of the street canyon?
Answer:
Yes, thank you for this. We will change this to view factors instead.
Equation 16 If psi_g is SVF at ground level then psi_v is SVF at...? Is this for ground-level vegetation (grass?) or tree canopy? or... This is not to clear to me, and for tree canopy things get more complicated? Maybe some more description would help the reader here and/or given an example for vegetation.
Answer.
Thank you for this. Basically, the SVF calculations for SOLWEIG differentiate between buildings and vegetation (canopy). We will clarify this.
Line 181 I was initially surprised at the very small distance of the IRR (infrared radiometer) from the surface, however Figure 1 is helpful for putting this small distance in context - so perhaps adding some words about the need for a simple lightweight mount attached to the surface of interest might be useful.
Answer:
The reason for the short distance is related to your following comment, where we elaborate a bit more on this.
As a more general comment: The IFOV of the wall surface for the IRR would be pretty small with such a small distance from the surface - and presumably the angle of observation is set to try and avoid any shadow issues within the IFOV? I guess this helps with not having a long period of sunlit/shadow transition on the wall if that is an issue for this position; on the other hand a larger distance would get a larger spatial sample and reduce any impacts of the sensor on the surface itself. The wooden wall also shows some microscale structure from the edge of the boards – possibly these are on a slight angle and have some microshadows beneath them? Do you think this has any impact on the model vs observed relation? Is the angle of view of the IRR set to try to avoid any such effects?
Answer:
Nice observation! The reason we put the sensors on angle and so close to the walls is exactly for the reasons you are describing above. We wanted to avoid self-shadowing from the sensor and also avoid potential “micro-shadows”. Whether there are “micro-shadows” or not we do not know and of course there could be some impacts from these. We, however, expect them to be small, and when comparing the observations with our simulations, we are comparing the relatively small field of view of the sensors with 0.5 m wall pixels (voxels). We will add a sentence or two about this in the discussion.
Line 182 Have you tried any sensitivity testing on the choice of these thermal parameters?
Answer: This is a very good comment. We will add a sensitivity test to the manuscript to help us understand more about our model and parameterization. Hopefully this can also help users understand some of the limitations of the model, and how to set input parameters.
Line 185 I guess this is average shortwave radiation incident on the plane of the wall (not horizontal incident shortwave at the top of the canyon?) Probably should make this explicit.
Answer:
This is actually the average input horizontal shortwave radiation (forcing data) from the weather station as an indicator of what type of meteorological days that we are using to simulate the wall temperatures seen in the figure. We will clarify this.
Line 187 Re emissivity setting of the IRR - it may be worth checking on what assumptions it is making regarding the incident longwave radiation on the target , which may not always be true in urban environments.
Answer:
Yes, there is likely some error from longwave radiation reflected from the background. According to the manual it would be based on the reflectivity (1-ε) of the target, where ε is the emissivity of the target. For our case ε is likely ~0.95, meaning that reflectivity is ~5%. To be able to estimate the background radiation we would have had to measure it, which we unfortunately did not. We will mention this in our discussion.
Line 305-307 Some older models might also be included? e.g. the wall temperature comparison done for the TUF3d model (Krayenhoff & Voogt 2007) and Henon et al. 2012 with SOLENE? And there may be others too.
Answer:
Thank you for this. We will include these.
Line 325 I might describe Ts as 'influencing' the sensible heat flux, since there are probably larger scale processes occurring within the street canyon that may influence the extent and direction of sensible heat flux.
Answer:
We will clarify this.
Line 327-328 I agree and the success of relatively simple models for estimating 3d urban temperature in applications of urban thermal anisotropy also point to the dominance of radiation as a control on surface temperature.
Answer:
Happy to hear that you agree!
Line 368 We may know the range of parameters but isn't a challenge knowing how to assign such parameters to the variety of buildings in an urban area and not necessarily knowing the cross-sectional structure of the walls for all buildings? Presumably this leads to a level of uncertainty that would be greater than for a test case with a building where more precise information can be determined.
Answer: Yes, this will always be tricky. What we are trying to say, maybe, is that with more complex models you would have to assign parameters to different surfaces as well as their cross-sectional structure. From a user perspective, i.e. end-users of the SOLWEIG model, simple input data is key. The SOLWEIG model is used quite extensively in Sweden and abroad. One reason for this could be that it is relatively simple to use. We have added functionality to add different building types/surfaces in our ground cover scheme, meaning that a building could be e.g. 100 for wood, 101 for brick and 102 for concrete. In this way it is possible to run SOLWEIG with different building types. As we have tried to demonstrate, we have also used input parameters from look-up tables, something that a user could also do. But one will never know if simulated Ts is true unless it is compared to observations.
Wording/typographical suggestions
Line 20 suggest “and quantify how it influences”
Answer:
Will change.
Line 41 Odd phrasing? The spatial differences in urban air temperature are typically small in most environments by day? (Key element here is daytime - since at night time the spatial differences in Tair are larger and better linked to surface characteristics).
Answer:
Yes, “Nordic” was based on the reference Lau et al., 2015. We will remove Nordic.
Line 105 Suggest "The omega term (in eq. 1) ..." (just to avoid starting the sentence with the symbol)
Answer:
Will change.
Line 108 Suggest: "The absorbed shortwave radiation" (the equations are representing the absorbed, not just the incident shortwave radiation)
Answer:
Will change.
Line 121 Suggest lower case w for where
Answer:
Will change.
Line 134 I think it would be useful to be explicit on what Lup and Ldown are.
Answer:
Will include this.
Line 135 Suggest: "of sunlit non-sky"
Answer:
Will change.
Line 147 Here an upper case version of the symbol is used compared to equations 5 and 10; suggest it be changed to match.
Answer:
Will change.
Lines 171-174 Perhaps this sentence should be moved to section 2.5?
Answer:
Line 178 Specify the wall aspect explicitly here? Looks to be south-east based on Figure 1.
Answer:
Yes, good, will do!
Line 222 suggest “and underestimates by 2-3°C in nighttime.”
Answer:
Will change.
Line 223 I'm not sure i understand the wording - do you mean they overestimate the surface temperature during times of intense solar radiation?
Answer:
Yes. Will change.
Line 233 I would move this sentence to the previous sentence that begins on line 230.
Answer:
Will move this sentence.
Line 239 suggest “are subject to partly cloudy weather”
Answer:
Will change.
Fig 4 The label of the regression line omits the + sign of the equation; a reader would presumably infer this but might be useful to give all the signs given it is written as an equation.
Answer:
We will add the + sign.
Line 316-322 Move this to the methods?
Answer:
We see what you mean here. We will have a look at this and see if we can move it.
Citation: https://doi.org/10.5194/egusphere-2025-2093-AC2
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AC2: 'Reply on RC1', Nils Wallenberg, 26 Nov 2025
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RC2: 'Comment on egusphere-2025-2093 (review)', Anonymous Referee #2, 30 Oct 2025
see attached pdf-file "review egusphere-2025-2093"
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AC1: 'Reply on RC2', Nils Wallenberg, 26 Nov 2025
Page 2, line 39: PET was first introduced in Mayer and Höppe (1987). Therefore, this citation should be added to Höppe (1999) or replace Höppe (1999). Höppe et al. (1999) is incorrect. Mayer, H. and Höppe, P.: Thermal comfort of man in different urban environments. Theor. Appl. Climatol. 38(1), 43-49, https://doi.org/10.1007/BF00866252, 1987. 2.
Answer: Thank you, we will of course add this reference.
Page 2, line 39: The quote about UTCI (Blazejczyk et al., 2010) is correct. However, based on the paper by Lee et al. (2025), it should be noted here that, for physical reasons, the UTCI cannot be applied in urban open spaces, i.e., within the urban canopy layer. Lee, H., Park, S., and Mayer, H.: Approach for the vertical wind speed profile implemented in the UTCI basics blocks UTCI applications at the urban pedestrian level. Int. J. Biometeorol. 69, 567-580, https://doi.org/10.1007/s00484-024-02835-x, 2025.
Answer: We will add this one as well and a sentence on the issue.
Citation: https://doi.org/10.5194/egusphere-2025-2093-AC1
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AC1: 'Reply on RC2', Nils Wallenberg, 26 Nov 2025
Data sets
SOLWEIG v2025 test dataset N. Wallenberg et al. https://doi.org/10.5281/zenodo.15309444
Model code and software
SOLWEIG v2025 as part of UMEP-processing N. Wallenberg et al. https://doi.org/10.5281/zenodo.15309383
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- 1
General Comments
In this paper, a new parameterization scheme for the SOLWEIG model is introduced and tested for a dataset from two walls with similar aspect and albedo, but different thermal properties. The evaluation shows good performance for the wall temperatures and a demonstration is made of the impact on SOLWEIG-predicted MRT. Overall, I think the work provides the basis for a useful contribution and the introduction of the scheme will broaden the capabilities of the SOLWEIG model. The step heating approach chosen seems to be based in part on calculation efficiency, although tests are not made to demonstrate its performance in this respect and I thought there was scope to include some sensitivity testing of the choice of thermal parameters. More generally, it was not clear to me if there are any assumptions implicit in the step heating approach that might impact its ability to represent environmental temperature time series. The basis of that approach seems to be in the laboratory testing of thermal properties of materials where a step change in heat input can be applied, but this may be different from the type of input from environmental radiative forcings. I recall here the comparison of PDE and Force Restore approaches for surface temperature modeling in Johnson et al. 1991 Boundary Layer Meteorology 56, 275-294 that dealt with some similar issues.
Specific Comments
Line 23 Does the choice of walls with albedo = 0.5 make the test somewhat conservative for Ts, given Ts would be expected to be lower in this case (compared to a wall with lower albedo?) Perhaps this choice is related to expected MRT?
Line 55-57 The Fourier's law of diffusion is just for the conduction of heat - so a surface energy balance approach could potentially replace Fourier's law with an alternative -e.g. a force restore approach could be used to resolve conduction while still using a surface energy balance approach.
Line 62 And possibly also to the season or time period of that calibration.
Line 67 Thermal effussivity is equivalent to thermal admittance (see e.g. Oke et al. 2017 Urban Climates) - which may be familiar to readers with an urban climate background so it could be useful to point out this equivalence.
Line 76-79. I wonder if a bit more context is needed - e.g. SOLWEIG is not quite a full energy balance model - and success at modeling Ts will also be related in part to the performance of the convective and conductive parameterizations (as well as radiation).
Line 84 Are there any issues given that the environmental forcing is different from a Dirac heat pulse as assumed in the reference?
Equation 1: Equation 1) has surface temperature on both sides of the equation, since outgoing longwave that is part of omega is determined by Ts – this may be worth noting.
Line 102 The matching is probably best for walls a few hours after sunset (e.g. to allow west-facing walls that may have been directly irradiated to cool closer to Tair)
Equation 13 I guess fsun could also depend on the canyon geometry? and some of the empiricism is related to expected obstructions within a canyon and/or microscale facet structure such as awnings that creates self shading?
Equation 15 I'm not sure why the "fraction" terminology is used when the other values are view factors? Essentially you are calculating the view factor of the opposite wall of the street canyon?
Equation 16 If psi_g is SVF at ground level then psi_v is SVF at...? Is this for ground-level vegetation (grass?) or tree canopy? or... This is not to clear to me, and for tree canopy things get more complicated? Maybe some more description would help the reader here and/or given an example for vegetation.
Line 181 I was initially surprised at the very small distance of the IRR (infrared radiometer) from the surface, however Figure 1 is helpful for putting this small distance in context - so perhaps adding some words about the need for a simple lightweight mount attached to the surface of interest might be useful.
As a more general comment: The IFOV of the wall surface for the IRR would be pretty small with such a small distance from the surface - and presumably the angle of observation is set to try and avoid any shadow issues within the IFOV? I guess this helps with not having a long period of sunlit/shadow transition on the wall if that is an issue for this position; on the other hand a larger distance would get a larger spatial sample and reduce any impacts of the sensor on the surface itself. The wooden wall also shows some microscale structure from the edge of the boards – possibly these are on a slight angle and have some microshadows beneath them? Do you think this has any impact on the model vs observed relation? Is the angle of view of the IRR set to try to avoid any such effects?
Line 182 Have you tried any sensitivity testing on the choice of these thermal parameters?
Line 185 I guess this is average shortwave radiation incident on the plane of the wall (not horizontal incident shortwave at the top of the canyon?) Probably should make this explicit.
Line 187 Re emissivity setting of the IRR - it may be worth checking on what assumptions it is making regarding the incident longwave radiation on the target , which may not always be true in urban environments.
Line 305-307 Some older models might also be included? e.g. the wall temperature comparison done for the TUF3d model (Krayenhoff & Voogt 2007) and Henon et al. 2012 with SOLENE? And there may be others too.
Line 325 I might describe Ts as 'influencing' the sensible heat flux, since there are probably larger scale processes occurring within the street canyon that may influence the extent and direction of sensible heat flux.
Line 327-328 I agree and the success of relatively simple models for estimating 3d urban temperature in applications of urban thermal anisotropy also point to the dominance of radiation as a control on surface temperature.
Line 368 We may know the range of parameters but isn't a challenge knowing how to assign such parameters to the variety of buildings in an urban area and not necessarily knowing the cross-sectional structure of the walls for all buildings? Presumably this leads to a level of uncertainty that would be greater than for a test case with a building where more precise information can be determined.
Wording/typographical suggestions
Line 20 suggest “and quantify how it influences”
Line 41 Odd phrasing? The spatial differences in urban air temperature are typically small in most environments by day? (Key element here is daytime - since at night time the spatial differences in Tair are larger and better linked to surface characteristics).
Line 105 Suggest "The omega term (in eq. 1) ..." (just to avoid starting the sentence with the symbol)
Line 108 Suggest: "The absorbed shortwave radiation" (the equations are representing the absorbed, not just the incident shortwave radiation)
Line 121 Suggest lower case w for where
Line 134 I think it would be useful to be explicit on what Lup and Ldown are.
Line 135 Suggest: "of sunlit non-sky"
Line 147 Here an upper case version of the symbol is used compared to equations 5 and 10; suggest it be changed to match.
Lines 171-174 Perhaps this sentence should be moved to section 2.5?
Line 178 Specify the wall aspect explicitly here? Looks to be south-east based on Figure 1.
Line 222 suggest “and underestimates by 2-3°C in nighttime.”
Line 223 I'm not sure i understand the wording - do you mean they overestimate the surface temperature during times of intense solar radiation?
Line 233 I would move this sentence to the previous sentence that begins on line 230.
Line 239 suggest “are subject to partly cloudy weather”
Fig 4 The label of the regression line omits the + sign of the equation; a reader would presumably infer this but might be useful to give all the signs given it is written as an equation.
Line 316-322 Move this to the methods?