Simulating satellite urban land surface temperatures: sensitivity to sensor view angle and assumed landscape complexity

Urban land surface temperature (LST) from satellite earth observation (EO) varies with sensor view angle. Where these variations are not accounted for, urban LST products are inconsistent through time, limiting their use in urban weather and climate model evaluations and process studies (e.g. urban heat island, building energy balance, human thermal comfort). Obstacle-resolving numerical models (ORM) of urban form and radiation exchanges are being used to: (a) understand relations between EO view angle, the 3-dimensional urban surface, and the surface temperatures from urban land surface models, and (b) evaluate parameterisations (parametric models) that aim to account for LST angular effects for the next operational satellite products. Most ORM are limited to simplified buildings (e.g. cuboids) and surface temperatures by lack of datasets. Novelly, we use both a realistic urban form model and observed surface temperatures to assess the impact of simplifying the urban form and temperature on the modelled LST anisotropy. We test various sets of assumptions in central London by combining ground-based thermal camera observations and the state-of-the-art Discrete Anisotropic Radiative Transfer (DART) model. The high resolution realistic model ( < 1 m) includes thermal (surface temperatures varying by sun-surface geometry, shadow history and materials every 30 min) and geometry (sloped roofs, chimneys and vegetation) data. These data are used to simulate brightness temperatures of EO pixels to quantify LST view angle variations. During daytime, a change of view angle of 47 ◦ off-nadir corresponds to a difference in LST of up to 5.1 K for the realistic building model. The intermediate-complexity landscape (easily obtainable building geometry/ footprints and more idealised surface temperature distributions driven only by shadow patterns) gives the best agreement in simulated LST to the realistic landscape. The directional variations are still captured in total (daytime mean absolute error 0.44 K) when using an idealised ORM representation of the same landscape (cuboid buildings, simplified surface temperature) except for roofs which are near-isotropic. Results suggest that geometry assumptions used in current ORM are suited for evaluation of parametric models used to develop and verify operational LST sensor view angle corrections. Future work should consider more realistic materials and scattering processes including low emissivity glass and metals with challenging specular properties.


Introduction
Surface temperature is a central variable in the surface energy balance.Satellite-derived land surface temperature (LST) can provide a spatially continuous, global product that has been used to map surface urban heat islands (Yang H et al., 2020;Miles and Esau, 2020) and urban air temperature (Venter et al., 2020) for an improved understanding of the effects of cities on the climate and heat stress of populations.Urban land surface models increasingly use LST for model evaluation (Giannaros et al., 2013;Hu et al., 2014).However, any remote sensing (RS) observation is limited to a two-dimensional view of the surface-leaving radiance to determine LST (Jiang et al., 2018).Urban three-dimensional (3D) morphology and complex surface materials create directionally variable (i.e.anisotropic) surface-leaving radiance (Krayenhoff and Voogt, 2016).Surface-leaving radiance sampled by a remote sensor can therefore change depending on the view angle.For example, an observation viewing directly downward (i.e.nadir) is biased towards the surface-leaving radiance from horizontal surfaces (Roth et al., 1989;Voogt and Oke, 1997;Hu and Wendel, 2019), missing the large wall area contributing to the "complete" surface temperature across all surfaces (Jiang et al., 2018;Yang J et al., 2020).Anisotropy of surface-leaving radiance across urban canopies is defined as an "effective" thermal anisotropy, to differentiate from the thermal anisotropy exhibited by individual surface components (Voogt and Oke, 1998a) caused by the non-Lambertian properties of real-world surfaces, including specular reflections from glass and metals.Effective thermal anisotropy in urban areas can give rise to directional variations in LST of over 10 K (Lagouarde et al., 2010) and is unaccounted for in current LST retrievals.This can confound estimates of surface urban heat island intensity through time and is considered a major source of model evaluation inconsistency between observed and modelled LST (Hu et al., 2014), along with cloud contamination.Further, it reduces LST climate record consistency, which is crucial if LST is to be considered a viable essential climate indicator or variable (WMO, 2016;ESA, 2019) or to be used routinely in data assimilation.
Future operational LST retrievals will convert angular LST to nadir or hemispherical temperature products by including models which normalise the angular variations (Cao et al., 2019) or "complete" surface temperature weighted across the full urban envelope (Voogt and Oke, 1997;Jiang et al., 2018;Yang J et al., 2020).The proposed models are semi-empirical parametric models (PM) that are computationally efficient and do not rely on complex inputs of land surface type, structure or component temperature distribution (Cao et al., 2019).PM describe the underlying processes of angular LST using kernels for (e.g.) emissivity, sun angle and shadow distributions.These are analytically expressed using multi-angular LST datasets, e.g.overlapping GOES-E and GOES-W measurements (Vinnikov et al., 2012) or combined geostationary and polar orbiters (e.g.Jiménez et al., 2012).Recently proposed PMs consider urban building/road geometry (Wang et al., 2020) and urban vegetation geometry (Cao et al., 2021;Jiang et al., 2021).
Using concurrent multi-angle LST for implementation of PM presents challenges.Observations from different satellites have spatial and temporal mismatches and have systematic product/instrumentation uncertainties (Ermida et al., 2014).Only ASTR and SLSTR sensors operate with two observation angles (Liu et al., 2020).Even differences in capture time for a multi-angle and multi-instrument observations (e.g.~10 min) can have an effect as urban facets exhibit large fluctuations in surface temperature at short time scales (Christen et al., 2012).Quantifying these uncertainties requires detailed evaluation from highresolution observations and simulations.3D models (hereafter "3DM") or obstacle resolving models (ORM) offer a detailed, physical basis to evaluate PMs and multi-angle LST through explicit definition of the observed surface geometry and temperature distribution.However, available ORM approaches also vary in terms of building complexity and representation of real-world surface temperature variability.For example, Jiang et al. (2018) evaluate the PM of Vinnikov et al. (2012) and simulate scenarios when multi-angular LST were unavailable using an ORM (CoMSTIR) with idealised surface temperatures and cuboid buildings.Wang et al. (2020) evaluated the GUTA PM using the DART ORM (Wang et al., 2018) configured with repeating idealised buildings.It is the overarching objective of this study to quantify the implications of assumptions during the ORM setup.This is vital so that ORM can then be used to assess uncertainties inherent in PM used to support the next generation of LST retrievals.

Abbreviations
A range of urban characteristics have not yet been addressed by previous thermal anisotropy modelling applications: i.With cuboid buildings and flat surfaces, sub-facet scale (e.g.< 1 m) shadowing is not accounted for.Sub-facet shadows can reduce a wall's mean surface temperature by 1-7 K (Hilland and Voogt, 2020), whereas roof shape (pitch and structure) can cause variations of over 20 K (Morrison et al., 2021).Prior ORM studies of LST anisotropy without micro-scale 3D structure hence tend to underestimate the anisotropy (Lagouarde et al., 2010;Krayenhoff and Voogt, 2016).ii.Urban vegetation is often a large fraction of the urban surface (Small and Lu, 2006) and generally much cooler than impervious surfaces (Meier and Scherer, 2012).To our knowledge only Dyce and Voogt (2018) have used ORM with urban vegetation to simulate directional LST, but using plane-parallel building and vegetation canopies (see i.).iii.Surface temperatures from ORM are very sensitive to model parameters that describe the surface composition (e.g.material thickness, heat capacity, albedo, emissivity).Still, to date neither these parameters nor the ORM surface temperatures are extensively evaluated as the necessary observations are both challenging to obtain and have limited spatial and temporal footprints (Pigeon et al., 2008).Ground-based observations, such as those across a residential area (Adderley et al., 2015), an idealised scale model (Morrison et al., 2018) or the New York city skyline (Ghandehari et al., 2018), do not cover the full complexity and diversity of realistic urban geometries.Although airborne observations (e.g.Sugawara and Takamura, 2006;Lagouarde and Irvine, 2008) and vehicle transects (Voogt and Oke, 1998a) have greater spatial extent, and hence represent higher complexities, they are snapshots in time.iv.Urban materials are generally unknown at high spatial (facet scale), spectral (short and longwave) and angular (e.g.specular reflectivity) resolution.Thus, ORM are typically prescribed with bulk and Lambertian optical properties.In stark contrast, real urban areas have extremely complex material distributions and spectral responses (Kotthaus et al., 2014), with plastic signatures (Guo and Li, 2020) and specularly reflecting glass and metal claddings (Kotthaus et al., 2014)."Cool" LST islands detected around tall buildings (Agathangelidis et al., 2020) may hence be anomalies associated with LST retrieval uncertainties caused by the influence of specific materials (e.g.high glass fraction) on the anisotropy (Crawford et al., 2018).
To assess such sources uncertainty in common ORM configurations and to quantify the relative impact of simplifying the urban form and temperature, respectively, this study uses a uniquely detailed model accounting for those characteristics (i.-iv.)commonly simplified elsewhere.This allows us to model urban thermal anisotropy, with varying degrees of complexity in the underlying processes whereby the implications of various assumptions can be quantified.Realistic urban form and surface temperatures that are neither often nor easily available are combined to simulate a heterogeneous urban area.The micro-scale surface structure of 3D buildings is here resolved to individual pitched roofs, chimneys, balconies.Building facets, ground surfaces and vegetation are assigned component surface temperatures derived from highresolution thermal infrared camera observations.Using a state-of-the-art radiative transfer model, a "realistic baseline" simulation of urban anisotropy is performed that then serves as a refence to asses simplified model setups.The complex landscape is incrementally simplified to obtain descriptions similar to previous studies to and the ability to characterize angular LST variability is compared to the "realistic

Modelling satellite LST variation by view angle
The The overall T b EO range (i.e.maximum (max) to minimum (min)) for distribution of temperatures at a given view angle is (Krayenhoff and Voogt, 2016): (2)

Model landscape
In this study, the term "landscape" describes a 3D model of urban form and surface temperature.Here, the urban form includes the buildings, ground, and vegetation, overlaid with a 3D surface temperature field with 1 m × 1 m × 1 m resolution voxels (Appendix A).Each voxel holds a data point (here surface temperature) for a given volume.Landscapes are created (Section 2.3) and then implemented in DART (Section 2.1.2) to simulate T b EO .

DART radiative transfer model
DART (here version 5.7.5) is a computer model that simulates radiative transfer in heterogeneous 3D landscapes.Using a ray tracing approach, DART determines the emission and multiple scattering of individual longwave infrared (LWIR) rays across a landscape.The result can be a 2D image of a landscape's surface-leaving thermal radiance across a horizontal plane at the top of the landscape (the bottom of atmosphere -BOA) (Yin et al., 2015;Wang and Gastellu-Etchegorry, 2020) at a given wavelength (λ, μm), with (x, y) the image pixel indexed by surface component(s) i (e.g.roof and/or wall), ϕ and θ ( • ) the zenith and azimuth propagation angles, and Ω (sr) the solid angle.Hereafter for brevity, Ω and λ are no longer stated explicitly as simulations are performed at a single selected wavelength and use uniform solid angle.For each timestep, 116 2D images are computed for directions θ = 0 → 360 • .To reduce computation time, the number of image files, and the data stored, only images for φ < 50 • are simulated.
To determine a landscape's view angle variation in upwelling radiance, L surf (x, y, ϕ, θ, i) is resampled to obtain a landscape mean surfaceleaving radiance typical for EO resolutions:  2020) uses Google Earth Pro data (Google, 2019), simplified (LOD1) as extruded building footprints (Evans et al., 2011) and idealised (LOD0, schematic) using study area building height, number, and plan area (Table 1). ) at the simulation wavelength (λ, μm): Here λ = 11.02μm is used, corresponding to the centre of MODIS (Moderate Resolution Imaging Spectroradiometer) band 31.A split window (Zhengming Wan and Dozier, 1996) or other (e.g.Islam et al., 2017) LST retrieval would then use L surf at different bands and consider neither the view angle or facet (i) level radiative exchange processes.

DART vs. previous sensor view modelling of anisotropy
Previous sensor view modelling of thermal anisotropy to derive T b EO (ϕ, θ) has determined the weighted contribution from various surface types (e.g.sunlit/shaded wall, roof, ground) for different view angles (e.g.Lagouarde et al., 2010): where f i (ϕ, θ) is the view fraction of surface component i within the sensor field of view determined by a sensor view model (e.g.Soux et al., 2004), T b,i is the surface component brightness temperature from observations or energy balance modelling, and n i is the total number of surface components.Surface components act independently and are typically opaque; but have included tree crowns (Dyce and Voogt, 2018).DART surface components can instead have radiative contributions from anisotropic scattering, transmission through turbid media (e.g.vegetation) and the (cloud-free) sky and are not constrained to idealised geometry (cuboid buildings, plane-parallel surfaces, smooth facets, etc.), bulk facet-level surface temperatures, or Lambertian emission.

Study area and observation period
The central London (UK) 420 m × 420 m study area has a mix of vegetation, high-and low-rise structures (Fig. 1).The analysis uses ground-based surface temperature observations taken in the area (Morrison et al., 2020;Morrison et al., 2021) for a mainly clear-sky summer day (27th August 2017).
To assess how representative the study area is of central London and a single EO pixel (e.g.MODIS/Sentinel) at a resolution of ~1 km × 1 km, the morphometric and land cover characteristics are analysed for a ~ 4 km × 4 km area around the site (Fig. 2a).South and east of the site is the central business district.Both average building height and plan area index (PAI, fractional area covered with buildings) decrease northwards, while vegetation fraction increases in more residential areas.The study area PAI is similar to the surroundings (Fig. 2), but there is more vegetation (> 75th percentile of the extended area, Table 1 cf.Fig. 2b, c) associated with a park (Fig. 1b, centred at grid coordinates 284,500, 5,712,850).As earlier ground-based thermography studies were undertaken in less built-up areas, our site's median building height (13.5 m, Table 1) is greater than both (Adderley et al., 2015) area-weighted building height of 6.23 m and (Morrison et al., 2018) 1.5 m cubes.From visual inspection, the vegetation fraction is lower than in the (Meier and Scherer, 2012) study area.

Model landscapes of the study area
The study area surface geometry, vegetation and temperature are simulated to create a series of model landscapes (Table 2, more details in Appendix A).Digital surface models (DSM) are used to represent the 3D geometry of each landscape.Their resemblance to the "real world" are indicated by their level of detail (LOD), where LOD2 (Fig. 3) has the most realistic building features (e.g.sloped roofs, chimneys, and balconies) and LOD0 the most idealised.In LOD1, the building footprints Evans et al. (2011), Fig. 1a) are vertically extruded (using "3Dfier", (Commandeur, 2020) to create 3D buildings with flat roofs and walls.As building footprints are widely available (e.g.Microsoft's Open Street Map (Heris et al., 2020)) LOD1 models are relatively easy to produce and hence most commonly used.
In LOD0 (Fig. 3), cuboid buildings are used with random orientations constrained so each wall angle has the same weight in an effective thermal anisotropy (Wang et al., 2018) estimate.The baseline LOD0 has regularly spaced cubes of randomly assigned heights (informed by the LOD2's building height inter quartile range) with both the PAI and number of buildings from the LOD1 building footprint.As LOD0 requires the least input information and pre-processing, it could potentially parameterise urban geometry globally; although coarsely and less accurately than higher LOD datasets.A range of LOD0 geometries are derived (hereafter "LOD0 ensemble") to account for uncertainties in the Table 4 Surface properties used to both stratify surface temperature observations and allocate them to a 3D landscape.Surface temperatures variation method M21 (Morrison et al., 2021) and ILU (Table 2).Acronyms see list at start of paper.Note as a BRF is relative to the illumination of a flat surface, a vertical surface (e.g. a wall) facing the sun is illuminated more than the reference flat surface, hence a BRF ratio is >1.bulk parameters used (Table 3): PAI ±10% (Esch et al., 2020), building height ± 3 m (Frantz et al., 2021) and spacing of cubes randomly up to 25% from the regular spacing distance.
All DSMs are given a broadband emissivity of 0.93 based on observed mean impervious material values (Kotthaus et al., 2014).This is used by DART to simulate multiple scattering of longwave (λ = 11.02μm, Section 2.1.2) radiation between surface components.
The vegetation in LOD1 and LOD2 are both the same, whereas for LOD0 there is no vegetation (Fig. 3).The vegetation is based on location of tall vegetation (i.e., non-grass) which includes trees and shrubs (> 2 m above ground level), hereafter vegetation canopy elements (VCE, Appendix A).As the leaf area density (LAD) can vary with vegetation type and volume, we consider some different densities (Table 2) but for each case all VCE voxels are assumed to have the same characteristics.The 1 m 3 voxels are assumed to be turbid with a spherical angular distribution of leaves, constant LAD and "deciduous leaf" optical properties at 11 μm from the DART spectral database (leaf transmissivity = 0.0145, reflectance = 0.0195).

Model surface temperature
Ground-based observations on a mostly cloud-free day at 30 min resolution are used to capture spatial and temporal variations of T s that are known to vary over short distances (Lee et al., 2018;Aguerre et al, 2019;Morrison et al., 2021).Previous ground-truth of urban T s are often limited to single-point infrared thermometers (e.g.Mathew et al., 2018;Wang et al., 2018), whereas we use thermal infrared cameras to better resolve spatial variations in surface temperature (Christen et al., 2012;Adderley et al., 2015).Observed surface temperatures (Morrison et al., 2021) are assigned to all landscape surfaces giving either a detailed distribution ("M21", Table 2), or simply based on solar illumination ("ILU", Table 2).The observed surface temperatures are corrected for atmospheric emission and absorption, and surface reflections, with uncertainties as determined by Morrison et al. (2020).Each voxel (X = Y = Z = 1 m) is assigned one T s (T s 3D (X, Y, Z)) in both methods (Appendix B).
However, a voxel can contain more than one surface component; for example, a building corner may have both east and south facing walls, or a balcony a south, down and roof facing surfaces.In these cases T s 3D (X, Y, Z, i) is set to the mean T s (i) of all surface components involved.The surface temperature voxels are saved as a pre-processing product for input into DART, with the surface temperature distribution across the DSM triangles determined by the triangle-cell intersection.For M21, the thermal camera observations are stratified based on three surface properties i (T s (i)) (Table 4) that drive surface temperature variation in urban areas (Morrison et al., 2021).These include: for example, a pitched roof's T s can vary with orientation to the sun; materials with higher albedos are associated with lower surface temperature through the lower absorption of solar radiation; and the effects of thermal inertia (Aguerre et al., 2019) are stratified by a combination of facet type and time in shade rather than using overall aggregates (e.g.Voogt and Oke, 1998b;Morrison et al., 2018).For ILU, T s (i) is stratified using sunshine status, surface orientation, and material only (i.e.no shadow history, Table 4).
Following the GUTA PM (Wang et al., 2018) evaluation using DART, a surface component T s range is calculated for direct input to the model.The range of mean temperatures is defined by shaded ("lower") and (caption on next column) W. Morrison et al. sunlit ("upper") limits.The "upper" limit T s is prescribed in DART to any sunlit surface, and the "lower" T s to any shaded surface.This gives 3D variations based on sun angle, DSM, and vegetation.While M21 is the best estimate of a realistic T s , ILU is both more easily simulated as it requires fewer surface temperature components, and more representative of studies limited by observations availability (e.g.IRTs, thermocouples, Li and Li (2020)) or idealised 3DM with plane-parallel geometry.The ILU setup is already implemented in DART, meaning the landscape geometry can change without needing to manually pre-process and store the M20 landscape temperature in large 5D numerical arrays (i.e., 3D space, shadow history timestep, surface class).
As the brightness temperature observations for vegetation are unstratified by sun-surface geometry, all VCE are assigned the same T s .This is the observed air temperature measured 1 km south east of the study area (Morrison et al., 2020).

Results and discussion
First, the highest detail landscape simulation, informed by observations and 3D modelling, is analysed (Section 3.1).Based on this, the best estimate of 2D satellite observations are derived.The simulations are unprecedented both in terms of spatial resolution (< 1 m) and temporal continuity (30 min resolution for a day post-sunrise).
Second, this benchmark is used to study a comprehensive range of idealised landscapes (Section 3.2) based on similar assumptions used in previous simplified approaches to interpret effective thermal anisotropy.These simplified models are required to verify any future operational corrections (e.g. based on GUTA, Wang et al. ( 2020)) for effective thermal anisotropy.

Spatially resolved LST variations in response to satellite view angle
To quantify variations in LST in response to the view angle across, high spatial resolution directional brightness temperature images (T b EO (x, y, ϕ, θ)), a combination of LOD2 geometry and M21 surface temperature observations (Table 4) is used.Depending on the view angle, different surfaces are captured in the image (Fig. 4) which creates unique brightness temperature distributions.Roof, ground, and vegetation are viewed from the nadir direction.Off-nadir view angles also have wall surfaces visible, many with complex T b distributions.The sub- The colder vegetation creates T b spatial patterns that contrast with warmer impervious surfaces.The generally homogeneous T b (T s prescribed based on air temperature, Section 2.4) is due to the uniform leaf area density (LAD0.7,Table 2).

Temporally resolved thermal anisotropy
While airborne studies capture temporal snapshots of anisotropy, here the analysis is continuous at 30 min resolution between 2 h after sunrise (08:00) and 2 h after sunset (22:00), allowing for a time series evolution of anisotropy (Fig. 5) to be investigated.It reveals that at 08:00, when T s is similar between facet types (Appendix B), T b EO (ϕ, θ) is near-isotropic across all view angles up to ϕ = 47 • (Λ = 0.7 K, maximum anisotropy, Eq. 2).

Effect of landscape simplification on thermal anisotropy
To identify key inter-landscape differences in brightness temperature from an earth observation perspective (T b EO ) for the different landscape configurations (Fig. 3, Fig. 4) we discuss a subset (Fig. 6a) of the study area.The highest detail T b EO (LOD2 geometry and M21 T s 3D , Fig. 6b) has the greatest T b EO (x, y) variation (cf.Fig. 6c-e), particularly across roofs where sub-facet geometry (e.g.sloped roofs, chimneys) have diverse sun-surface geometries.
In Fig. 6b, M21's sun-surface geometry classification (BRF, Table 4) prescribes a range of T s .For example, T b EO (x, y) across the sunlit pitched roof (centred x = 25, y = 370) has around 307 K → 316 K T b EO (x, y)   difference between left (more east-facing) → right (more west-facing) sides.While the left side has a low sun illumination angle (low irradiance and BRF), the right side is almost perpendicular to the sun angle (high irradiance and BRF).The parameterisation ILU (Fig. 6c) has Boolean "sunlit" and "shaded" temperatures across T s 3D (Section 2.4).This results in the pitched roof being mostly homogeneous as it is prescribed "sunlit" roof T s .Across other roofs, T b EO (x, y) ILU varies only if there is self-shadowing from micro-scale roof features.ILU's lower micro-scale variation in roof temperature results in a T b EO (ϕ = 0 • ) (308.0 K) that is 2.5 K higher than for the M21 (305.5 K) case.Similar micro-scale effects have been reported for walls with ~20-40% self-shading (Hilland and Voogt, 2020).With flat roofs (LOD1) there is no micro-scale T s variation (Fig. 6d).LOD1's buildings are generally more pronounced given their flat facets with sharp right angles when roofs and walls intersect.Some buildings  3).
W. Morrison et al. are present in LOD1 but not in LOD2 (and vice versa) because of differing DSM data sources.LOD1, created around 2010, has buildings at x = 50, y = 390 and x = 0, y = 420 (Fig. 6) which are absent in LOD2 (created 2018).However, manual inspection across the study area, suggests the LOD1 footprints generally agree well with the LOD2 buildings (Fig. 1a).
For vegetation, both the geometry and temperature descriptions in the LOD1 and LOD2 simulations are the same.Hence, the VCE have homogeneous T s .However near the canopy edges, T b EO (x, y) varies where the canopy is often thinner (e.g.Fig. 6b) allowing radiation from the warmer ground below to penetrate.Although, the LOD0 cubes (Fig. 6e) have no micro-scale T s variation and VCE are omitted, the T s responds to the shadows cast on the ground creating the coolest regions.The cube roofs are warmest.The VCE are not included as the layout relative to real-world buildings (cf.cubes) is not matched.Using idealised cuboid vegetation to test the sensitivity of various vegetation and cube morphologies to thermal anisotropy (see Dyce and Voogt, 2018) is outside of this study's aims.

Surface temperature distribution
The two different surface temperature distributions (M20, ILU, Table 2, Table 4) are compared to understand the contribution of shadow history and surface cooling through time impact on T b EO (ϕ, θ) variations.T b EO (ϕ, θ) is simulated using the simplified temperature across the landscape (T s 3D =ILU, Table 2) and compared to results using the complex distribution (T s 3D = M21; Table 2, Section 3.1).
As anisotropy from surfaces' both view fractions (f i (ϕ, θ), Appendix C) and temperatures (T s,i , Eq. 5) drives ΔT b EO , these are explored to understand the differences in anisotropy in response to LOD (Fig. 8).LOD2 has the smallest roof view fraction (f i=roof , Fig. 9g) as pitched roofs and awnings (etc.) with a deviation of >45 • from horizontal are classified as a wall (Appendix A), with the lower roof fractions offset by higher fractions of wall.LOD2 f i=roof is the most anisotropic, decreasing from 0.31 at nadir to 0.27 for a view of south-facing surfaces (ϕ = 47.1 • , θ = 214.1 • ; Fig. 9g).At nadir the tops of any convoluted roofs are most visible, but with increasing view angles lower roof structures can be occluded by other roof features.Comparing across all azimuth angles at ϕ = 47.1 • , LOD2 f i=roof varies between 0.27 → 0.32.LOD1 and LOD0 f i=roof are near-isotropic (~0.02 range, Fig. 9h, i), consistent with assumptions made in most parametric models and idealised 3DM.This makes f i=roof approximately same as the plan area index for any direction (Hu and Wendel, 2019;Wang and Chen, 2019).f i=- ground is generally consistent across LODs (Fig. 9d-f).
LOD2 roofs have a cool bias of over 5 K around midday.This is explained by high albedo "cool" surfaces (Appendix B, ~17.9% of total roof area, Morrison et al. (2021)) along with micro-scale geometry effects of self-shadowing, particularly from sloped surfaces towards/away from the sun (e.g.compare Fig. 6 b, d).
For wall and ground components, all LODs have directional brightness temperature variations (Fig. 10a) as for each view direction different shaded/sunlit fractions are seen.Brightness temperatures for LOD2 ground are lowerparticularly at nightas LOD2 has a 0.17 grass fraction (Table 1) which has a relatively low surface temperature (Fig. B-1), whereas in the other LOD grass surfaces are unaccounted for.There is some unexplained azimuth variation in LOD1 wall and ground view fractions (Fig. 9b, e) perhaps from a bias in certain building orientations which, in central London, would be reduced when using a larger domain area containing more streets and buildings.
The combined anisotropy from walls and ground of LOD2 (Fig. 10b, blue) is similar to that of LOD1 and LOD0 (MAE LOD1 = 0.15, MAE LOD0 = 0.26 K) and thus contributes less to the overall inter-LOD ΔT b EO differences (Fig. 8).The generally high correlation between all landscapes (Fig. 8) is further understood through analysis of inter-landscape view fractions (Fig. 9) and facet temperatures (Fig. 10) which combined (e.g.Eq. ( 5) give rise to the observed anisotropy.

Vegetation
To investigate the impact of vegetation density on daytime anisotropy (Fig. 11

Conclusions
Urban geometry and surface temperature data are used to simulate directional variations of satellite brightness temperature images -the effective thermal anisotropy.Three-dimensional (3D) modelling of urban form and thermal radiative transfer processes using the Discrete Anisotropic Radiative Transfer (DART) model are used to explore the W. Morrison et al. relative importance of the geometry description, surface temperature representation and presence of vegetation.The study demonstrates these processes for a mainly clear-sky summer day in central London when high-resolution measurements were available.
For the first time, modelling of effective thermal anisotropy is not constrained to idealised descriptions of urban form and temperature.A realistic geometry with sub-facet features (e.g.pitched roofs and 3D vegetation canopies) is used as a reference standard to assess implications of using highly idealised urban geometry descriptions, such as repeating cube buildings, similar to what may be assumed in operational corrections of effective thermal anisotropy.
The 420 m × 420 m study area is similar to the larger central city surroundings (3 km × 3 km around the study area), based on morphometric parameters (e.g.building height, plan area) variability (defined here by interquartile range), except for tree cover (17%) that is within the 75th -80th percentile.If similar detailed 3D model information and very high-resolution surface temperature data were available for a wider range of urban forms, the current simulations and analyses could be extended to capture a more representative diversity of intra-and intercity variations in thermal anisotropy.However, such data do not yet exist for extensive areas.
Simulations of satellite, or Earth Observation (EO), views of the surface are used to quantify the effective anisotropy, defined as the difference in brightness temperature (T b ) between a view direction that is off-nadir (oblique) and a nadir (downward) view direction.The simulations use classified surface temperatures observations, whose allocation includes shadow state obtained from simulations considering either (1) known histories, or (2) instantaneous patterns.Although the anisotropy "hot spot" is generally broader if shadow history throughout the day is accounted for, both have close daytime agreement (MAE = 0.2 K).This allows the conclusion that instantaneous shadow patterns are a reasonable simplification when assigning surface temperatures in this urban setting in DART.
With instantaneous shadows as the basis to assign the facet temperature distributions, the role of the level of detail (LOD) of the building geometry on anisotropy is analysed based on three cases: realistic buildings (LOD2), planar roofs/walls (LOD1) and repeating cubes (LOD0).Consistent with earlier studies, anisotropy increases with increasing morphology detail.Simulated anisotropy using LOD1 is in closer agreement (daytime MAE = 0.16 K) to LOD2 than when using LOD0 (daytime MAE = 0.44 K).
The contribution of each facet type (roof, wall, ground) to the effective anisotropy is quantified for the three LOD.Changes in roof geometry complexity between LOD2 and LOD1 become apparent but in general both LOD lead to similar directional variations in the fraction of wall, roof and ground "seen" by the satellite (view fraction).LOD2 roof temperatures are generally cooler (on average 5 K cooler around midday) because of micro-scale shadowing.This micro-scale shadowing cool bias is the same order of magnitude as prior observational analysis.LOD2 results have more anisotropy than the flat LOD1 (and LOD0) roofs.Daytime directional brightness temperatures for the LOD2 roofs have a maximum 3.4 K range (14:00) compared to 1.5 K (0.2 K) for LOD1 (LOD0) (also at 14:00).The thermal anisotropy for the combined wall and ground facets are more consistent between LODs.
Including vegetation (trees, shrubs taller than 2 m) reduces the anisotropy for the daytime hot spot directions by around 0.4-0.6K.
Anisotropy is generally insensitive to variation in leaf area density, but future work should use the 3D radiative transfer modelling capabilities to investigate additional vegetation parameters (sunlit and shaded distributions, optical properties, leaf angular distributions, tree trunks and branches).Overall, vegetation is less important than LOD variation.In this study, vegetation (median height = 10.01 m) is generally shorter than the buildings (median height = 13.49m).Where vegetation frequently exceeds building heights (e.g.some residential settings) it may have greater implications on the effective thermal anisotropy because it would affect the roof fraction "seen" by the satellite.
Surface temperatures are prescribed based on several micrometeorological processes (solar illumination, shadow history, materials).Future studies should include broader spatial extents from airborne measurements that are spatially and temporally filled with facet-scale surface temperatures from energy balance models.Exploring the relative importance of geometry under different weather and climate conditions requires more observations.For example, the sensitivity of roof geometry to sun angle may differ during low winter sun angles.
This work has implications for how satellite based EO surface temperatures should be interpreted within urban areas and how EO data could be used in data assimilation and model evaluation in the future.Modelled thermal anisotropy using the easily derived LOD0 data are shown to give rapid and reasonable estimates compared to the more realistic cases.However, care should be taken using these lower LOD urban forms without prior evaluation at higher LOD.Expansion of the methods across different geometry and material emissivity configurations could test if the idealised LOD0 cases are applicable globally.There is scope for improving the material description of the realistic LOD2 cases, using increasingly available surface emissivity datasets through airborne and high-resolution satellite products.Implications of using idealised bulk emissivities can be explored, given highly reflective and specular surfaces common in cities (e.g.metals, glass) are expected to play a role in LST view angle sensitivity.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.4), according to sun-surface geometry and time in shade and their mean (green points), with the ambient or "well shaded" surface temperature (red points) used in M21 and ILU and the ILU sunlit temperature (blue points) for bidirectional reflectance factor (BRF) of 1 (i.e.surface temperature representative of horizontal surfaces).(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

B.2. Gap-filling surface component temperatures
As observations do not cover all possible facets of the entire study area (i.e. more surface component permutations exist in 3D than are observed) gap-filling is required.With no downward facing surfaces (e.g.underside of balconies) sampled by any camera, 'all shaded walls' observations are used for this class of per-pixel gap-filling.Further gap-filling details of the observed surface temperature are given in (Morrison et al., 2021).

Appendix C. Calculation of surface component radiance and view fractions
The fraction of surface type i seen within an image (f i ) impacts directional variation in urban LST.If f i for a nadir viewed urban surface has a plan area index of 0.5 (i.e.f i=Roof = 0.5) and the remainder is ground, then f i=Ground = 0.5, (i.e.all seen types equal 1).These values and components will change with view angle.
The simplest method to determine f i for a given view angle (f i (ϕ, θ)) is to sum the number of pixels per surface class i (e.g.L surf (x, y, ϕ, θ, i)) and calculate a fraction relative to the total image pixels.However, this is inaccurate if an image pixel has surface-leaving radiance contributions from more than one surface component: the view fraction can be double-counted.Approaches to address this include: sensor view "sub-patches" (Krayenhoff and Voogt, 2016) assuming all components of a "mixed" pixel has the same temperature; and sensitivity tests to a "counted" f i to the resolution of surface components derived from simulated RGB renders (Lagouarde et al., 2010).
Here we adopt a radiative transfer and energy conservation approach to calculation of f i , using DART simulations.DART radiance images are calculated per surface component (L i surf (x, y, ϕ, θ)) and have surface-leaving radiance contribution after all scattering iterations.Whereas, L surf (Eq. 3) W. Morrison et al. will always have the combined radiance contribution from all surface components.L i surf (ϕ, θ) is related to L surf (ϕ, θ) by: L surf i (ϕ, θ, x, y) (C-1) i.e. the mean across all (n px ) L i surf image pixels is L i surf (ϕ, θ) (Eq.C-1) and the sum of all (n img ) L i surf images is L surf (ϕ, θ) (Eq.C-2).L i surf pixels without surface component i have L i surf (x, y, ϕ, θ) = 0. L i surf pixels containing only a fraction of surface component i will have the associated fractional radiance contribution, which is more accurate than the aforementioned "simple approach" that resolves pixel-level view fractions.
A DART simulation for each landscape LOD is run in the thermal domain (11 μm) with a known isothermal temperature across all surfaces (T s 3D = 300 K).All L surf images have radiance B − 1 (300 K, λ = 11 μm) = 9.57W m − 2 sr − 1 μm − 1 .With this total radiance contribution known and held constant, L i surf (x, y) is then a fraction of L surf (x, y): f i (x, y, ϕ, θ) = L surf i (x, y, ϕ, θ) / ( L surf (x, y, ϕ, θ) = 9.57 ) . (C-3) Across a whole image with n px pixels, the view fraction of a surface component f i (ϕ, θ) is: and the sum of n surface component fractions is unity: The post-processing of these DART simulations is done in R using the daRt package (Morrison and Benjamin, 2020).

Fig. 1 .
Fig. 1.Study area: (a) height of all surfaces above sea level (asl) from Google Earth imagery (Morrison et al., 2020) with building footprints (black lines, from Evans et al. (2011)), (b) orthorectified RGB image from a mosaic of Google Earth (Google, 2019) images (c) looking south east from 76 m roof centre of scene.Coordinate Reference System WGS84 UTM grid zone 31 N.

Fig. 2 .
Fig. 2. Variability of surface characteristics in central London between grid-cells (n = 81, each 420 m × 420 m) compared to the study area (red) with (upper) maps and (lower) histograms with n th percentiles (P n ) of P 25 → P 75 (dashed lines), P 50 (solid line), with solid red line the study area value.(a) Land cover (4 m resolution) with extent of maps (white dashed box) in bl and a sample MODIS M*D11A1 pixel (orange box).Surface characteristics: (b) plan area fraction of tall vegetation (> 2 m) (c) plan area fraction of grass (d) plan area fraction of buildings (i.e. plan area index), (eh) building height, (il) sky view factor(Dozier and Frew, 2002) calculated in R (R Core Team, 2017) with the "horizon" package (Van Doninck, 2018), and (g, l) standard deviation of the respective parameter.Maps oriented to grid north (WGS84 UTM grid zone 31 N).(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4 .
Fig. 4. Study area surface-leaving radiance shown as brightness temperatures (T b ) simulated using three-dimensional (3D) surface temperature derived from observations and the DART model for 27th August 2017 at (left) 11:00 and (right) 14:00 for (top) nadir and (middlebottom) off-nadir zenith (ϕ) angles.Black dashed box (top left) is the area shown in Fig. 6a-d.Images have orthographic sensor perspective projection with surface → sky view directions such that: θ = 0 • is grid north and θ = 90 • grid east (WGS84 UTM grid zone 31 N) and ϕ = 0 • (ϕ = 90 • ) is viewing directly upward from (parallel to) the surface.White areas are the view inside buildings at the edge of the domain and masked from subsequent analysis.

Fig. 9 .
Fig. 9. View fraction f (Eq.C-4) of surface component (i) across simulated satellite perspective for (a-c) f i=Wall , (d-f) f i=Ground , and (g-i) f i=Roof .Satellite perspectives simulated are for zenith angles (ϕ) 0 • → 47 • (central points are ϕ = 0 • , black radial line grid spacing of 10 • ) and azimuth angles (θ) 0 • → 360 • with (colours, polar plots) the view fraction for a given direction.Scatterplots (insets) compare all directions between LOD2 (x-axis) and LODN (y axis) with (red) 1:1 line.Landscapes based on different level of detail (LOD, see Fig. 3 for definitions): (a,d,g) LOD2 -most realistic urban geometry, (b,e,h) LOD1 -realistic building footprints, and (c,f,i) LOD0 idealised cube buildings.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 10 .
Fig. 10.Simulated satellite brightness temperature (T b EO ) for different surface types i (e.g.i = Wall, i = Roof) for urban landscapes with different geometry level of detail (LOD) and surface temperatures (prescribed based on sun illumination, ILU) across view angles (ϕ, θ) up to 47 • off-nadir showing (a) mean T b EO (ϕ, θ, i) and range (vertical lines) during the day (x-axis, UTC) for idealised cubes (LOD0) to realistic geometry (LOD2); and (b) contribution of surface components to anisotropy shown as [off nadirnadir] differences (ΔT b EO (i) Eq. (1).
) we compare a baseline ΔT b EO simulation with complex geometry but no vegetation (ΔT b EO (LOD = 2, LAD = 0)) to cases with varying leaf area density (LAD).Generally, the baseline ΔT b EO has the most variation through time and with view angle.Adding vegetation reduces the anisotropy (consistent with Dyce and Voogt (2018)) with greater reductions where ΔT b EO > 0 K (Fig. 11, xaxes).For these viewing directions, the landscapes with LAD >0 have ΔT b EO around 0.4-0.6K less than the baseline.Once vegetation is accounted for, there is little sensitivity to LAD (Fig. 11, colours) and LOD (Fig. 11, columns).

Fig. B- 1 .
Fig. B-1.Surface component temperatures used as input for the three-dimensional surface temperature (T s 3D) parameterisations with the range (vertical lines) of possible sunlit and recently shaded values that are used in the M21 parameterisation (Table4), according to sun-surface geometry and time in shade and their mean (green points), with the ambient or "well shaded" surface temperature (red points) used in M21 and ILU and the ILU sunlit temperature (blue points) for bidirectional reflectance factor (BRF) of 1 (i.e.surface temperature representative of horizontal surfaces).(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 3
Uncertainty range used in morphometric parameters for the lowest level of detail (LOD0) geometry description of the London study area.EO indicates earth observation techniques are used.± 10% > 90% accuracy [EO building masks in European cities] (Esch et al., 2020) Mean building height ± 3 m Mean bias error [EO in Germany] = 3 m (Frantz et al., 2021) Building spacing ± 25% Real-world cities have irregular building spacing (Chen et al., 2017) W. Morrison et al. with n the number of image pixels.L surf (ϕ, θ) then relates to T b EO (ϕ, θ) by inversion of the Planck function (B λ − 1