# Modal decomposition for the analysis of solar radiation variability in urban areas

Guillaume Le Gall, Martin Thebault, Cyril Caliot, Julien Ramousse

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Guillaume Le Gall, Martin Thebault, Cyril Caliot, Julien Ramousse. Modal decomposition for the analysis of solar radiation variability in urban areas. International Journal of Heat and Mass Transfer, 2025, 251, pp.127316. 10.1016/j.ijheatmasstransfer.2025.127316. hal-05124370

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Modal decomposition for the analysis of solar radiation variability in urban areas*

Guillaume Le Gall<sup>a,*</sup>, Martin Thebault<sup>a</sup>, Cyril Caliot<sup>b</sup>, Julien Ramousse<sup>a</sup>

<sup>a</sup>LOCIE UMR 5271, Université Savoie Mont Blanc, CNRS, 60 Avenue du Lac Léman, Le Bourget-du-Lac, 73370, France

<sup>b</sup>CNRS, UPPA, E2S, LMAP, 1 Allée du Parc Montaury, Anglet, 64600, France

**Abstract**

The morphological heterogeneity of three-dimensional structures largely contributes to causing significant variations in the solar radiative flux density received by a specific intra-urban region, resulting in a multi-dimensional signal expanding across a wide range of spatial and temporal scales. This paper presents a unique approach for analysing the spatiotemporal variability of the irradiance field in an urban context by means of modal decomposition. A parametric investigation is conducted on a set of mid-high latitude districts, defined as homogeneous arrangements of cuboids with grey opaque Lambertian materials. Changes in the buildings density (total site coverage $\kappa$) and average height (global aspect ratio $H^*$) are investigated. For each configuration, the annual shortwave radiative flux density on the envelope of the central building is obtained with a backwards Monte Carlo method under clear sky conditions. Simulated fields are preprocessed to account for urban-only effects and decomposed in their eigenbases of variation using Principal Component Analysis (PCA). Results demonstrate the potential of modal decomposition like PCA for apprehending the separate variability of urban irradiance in both space and time, paving the way for the classification of district regions prone to specific variations of the solar resource. Characteristic temporal frequency bands are represented by successive modes, pointing out annual and daily periods of high variability. Related spatial disruptions to the global irradiance field from urban geometries are portrayed, informing about dominant morphological specificities. Changes in the diffuse radiation component depicted by

*Fundings: This work was supported by the French National Research Agency under the ‘Invess-tissements d’Avenir’ [grant number ANR-18-EURE-0016 — Solar Academy].

*Corresponding author

Email address: guillaume.le-gall@univ-smb.fr (Guillaume Le Gall)
prevalent modes are further distinguished from the blocking of direct radiation by
urban obstacles portrayed by subsequent less-impactful eigencomponents.

***Keywords:*** Modal decomposition, Principal Component Analysis, Shortwave radiation, Spatiotemporal variability, Urban environment

# 1. Introduction

The radiative flux emitted by the sun restricts to a specific band of the electro-magnetic spectrum, being almost entirely contained in the effective range [0.3 µm; 3 µm], so that it is commonly referred to as *shortwave radiation*. This radiation is, by nature, inexhaustible and universal, setting sunlight as an abundant natural supply of energy available worldwide. With the view of moving towards a carbon-free electric grid, substantial efforts are being made to position solar energy as a serious long-term alternative to more conventional non-renewable fuels [14, 30]. With 60 to 80% of global energy needs being directly consumed on the metropolitan scale [35], urban areas have become apparent homes for the production of solar power. All single surfaces (e.g. façades, roofs, roads) would be soon capable of collecting radiation from the sun, allowing urban energy to be directly produced where it is eventually used.

## 1.1. Spatiotemporal variability of shortwave radiation

The usual inherent complexity of urban environments yet often restricts the implementation of solar harvesting systems in these areas [12] (Fig. A.1). The high occurrence of diverse obstacles like buildings, vegetation and other infrastructures creates an aggregation of countless singular three-dimensional geometries, naturally reducing the overall sky factor and considerably disrupting incident radiation [25]. The variety and heterogeneity of their respective morphological characteristics (e.g. shape, size, orientation and inclination) and material optical properties (e.g. specular and diffuse reflectivity, transmissivity) further adds on this modulation [6]. Irradiance, i.e. the amount of solar radiative flux received by a surface per unit area [W.m⁻²], indeed becomes unevenly distributed in urban areas, with complex dynamic overshadowing effects appearing between structural elements. Important changes may especially arise from harsh shadows cast on single surfaces by the obstruction of direct radiative beams from more-or-less distant features [1]. Solar rays reaching the urban fabric also endure multiple reflections and other radiative events with local structures. This contributes to the diffuse component of urban irradiance, which gains importance in intricate areas
and additionally alters the global illumination [1]. Urban light reflections also lessen
the aggregated city albedo due to a lower direct sunlight penetration compared to an
open rural area [6]. The role of building façades is particularly important here.

These interactions between solar rays and the urban fabric eventually add on their
prior alteration by the stochastic atmospheric layer and, to a wider extent, the natural
variability of solar irradiation (typical daily and seasonal cycles).

Overall, a plethora of miscellaneous variables shape the interaction of solar rays
with a specific urban surface by inducing important spatiotemporal changes in the re-
ceived irradiance. These changes can hold a deterministic (i.e. variations) or stochastic
(i.e. fluctuations) nature, fatally leading to a multivariate solar resource becoming hard
to forecast by simple models or from direct observation [13]. Its variability yet remains
to be characterised as it is assumed to be amongst the main barriers to the global ex-
pansion of solar energy within the next decades [24].

The variation of sunlight in space and time has been extensively studied on the
scale of large territories, such as regions or countries (mesoscale to macroscale). Mul-
tiple probability laws and prediction models have been developed [34]. Micro-climate
modelling also benefits from a wide knowledge on the scale of the building itself [25].
However, the spatiotemporal variability of solar radiation in the urban environment
itself, considered as a pool of infrastructures, has remained almost unexplored so far.
The complexity and variety of their resulting interactions mentioned hereinabove is a
prevailing reason here. Considering the three-dimensional aspect of urban elements
yet remains crucial, as it induces additional variations not estimated within mesoscale
and macroscale models. This fine and accurate characterisation of the urban solar
radiative flux variability is essential at many levels, starting with the definition of
reliable predictive models for climate change [44]. The intermittency of urban irradi-
ance, which could reach minutes to hour scales, makes its collection and subsequent
use particularly complicated. This is all the more striking in intricate environments
as increasingly observed in modern cities, and represents real challenges for network
managers, e.g. for estimating the solar power output and optimising the harvesting pro-
cess [1]. More generally, an exhaustive knowledge of the solar resource in urban areas
is essential for the integration, management and operation strategies to be reliable,
robust and performant. Thus, a massive deployment of solar energy on the city scale
is impossible if its intrinsic spatiotemporal variations are not correctly apprehended.
## 1.2. Modal decomposition of intricate multi-dimensional phenomena

This important diversity of elements encountered in urban environments causes the density of solar radiative flux received by a specific surface to be the result of multiple interactions between several degrees of freedom. In other words, the field of irradiance over a specific urban region constitutes a multi-dimensional signal depending on many parameters and expanding across a wide range of spatial and temporal scales. *Modal decomposition* methods can be used to appreciate the global dynamics of such phenomenon by extracting a subset of its dominant behavioural features. A lot of work and efforts have been put in the development of these algorithms, starting within the framework of Model Order Reduction (MOR) motivated by fast improvements in numerical capabilities [38]. MOR is deeply-rooted today, with a plethora of implementations in diverse disciplines and various objectives in mind [29, 40]. In multiple situations, algorithms involved in MOR have yet been applied with the sole objective of characterising the dominant variability patterns, or *modes*, of an intricate phenomenon as an attempt to gain insight into its global behaviour [17, 42]. The final outcome now becomes analysis-oriented, with the dimensionality reduction being primarily applied to interpret the physics ruling the signal dynamics.

Since the introduction of Fourier series in 1807 and first matrix reduction attempts by Lanczos [27], multiple statistical analysis methods for approximating intricate multi-dimensional datasets have emerged in various fields, being all credited with their own respective assets and drawbacks [38]. Among the variety of strategies employed for dimensionality reduction, Principal Component Analysis (PCA) [33, 21, 26] has been extensively used and validated in multiple areas, including renewable energies [13, 14]. Depending on the discipline, a different designation may be used, e.g. Principal Orthogonal Decomposition (POD) in computational fluid dynamics [2], Hotelling Transform [21] and Karhunen-Loève Expansion [32] in image processing or Empirical Orthogonal Functions (EOF) analysis in climate science [31].

PCA is an *a posteriori* algorithm, i.e. a data-driven technique allowing for a physics-informed analysis. It comes up with a set of new empirical features which are the most representative of its overall dynamics, namely observed principal directions in which data varies. The use of PCA, or equivalent modal decomposition algorithms, as an analysis tool to deal with the solar radiation variability has remained weakly explored so far. To the authors knowledge, most of the research in that context has focused on irradiation forecasting and estimation [8], often employed within a Machine
Learning (ML) framework [14, 11]. When applied in the very purpose of a variability analysis, large-scale approaches have been exclusively adopted, therefore overlooking detailed urban specificities [45]. So far, no attempts have thus been made to employ modal decomposition techniques for investigating the spatiotemporal variations of solar irradiance in an urban context, at an obstacle-resolving degree.

1.3. *Objectives and paper structure*

This paper proposes a novel approach for analysing the variability in space and
time of the shortwave radiative flux density distribution on the scale of the city by
means of modal decomposition. The present work builds on the methodology initi-
ated by [28] and explores its robustness to variations in common urban morphological
factors. To guarantee the rigour of the conducted investigation, holding all the more
importance when a physics-based interpretation of the input phenomenon is sought,
a simplified district model is assumed. This study indeed remains preliminary and
serves as a validating step for a subsequent application of the approach to more com-
plex real-like configurations. In that objective, a parametric investigation is conducted
on a set of theoretical squared urban districts, defined as homogeneous arrangements
of cuboid-like buildings with grey opaque Lambertian materials at a typical mid-high
latitude. Changes in the building density, via the total site coverage κ, and average
height, via the global aspect ratio H*, are investigated, as likely to have an apparent
impact on the spatiotemporal distribution of incident irradiance across the district.
Namely, four neighbourhoods with distinct couples {H*, κ} are defined and assessed
against a fifth reference case. For each configuration, the shortwave radiative flux
density is modelled at each point of a regular grid (1 m² spatial resolution) on the
envelope of the central building via backwards Monte Carlo ray tracing [5]. The sim-
ulation is performed under clear sky conditions over a full year, with a 15 minutes
timestep. Sections 2.1 and 2.2 detail the physical model and annual solar radiation
simulations, respectively. A modal decomposition of the simulated fields of irradiance
is subsequently performed using Principal Component Analysis (PCA) (Section 2.3).
Dominant modes of variability obtained for each district setup are then analysed and
discussed in Section 3. The underlying physics is interpreted to help the understand-
ing of spatiotemporal variations in the distribution of incident radiation induced by
local urban features. Limitations and future work (Section 3.5), as well as the pre-
processing stage of space-time fields fed to PCA (Section 3.2), are all further detailed
here. Conclusions are finally provided in Section 4.
## 2. Modal decomposition of shortwave radiation in a theoretical urban context

### 2.1. *Model geometry and radiative properties*

A theoretical urban district of identical buildings with cuboid shapes is considered (Fig. A.2). The effective ground surface is a symmetric plane of side $L_D$ surrounded by an additional $l_D$-width buffer band ($l_D = 0.1L_D$). All buildings have the same height $H$ and an $L_B$-length squared basis. The horizontal dimension $L_B$ is set here to $0.18L_D$, providing a shape factor$^f$ of 0.22. Buildings are regularly arranged around a central cuboid (displayed in green in Fig. A.2) with a street width $\Delta L_D$ in both $(O, x)$ and $(O, y)$ directions. For the sake of practicality, and to maintain consistency with actual proportions of typical european metropolitan areas, $L_D$ is set to 100 m.

A total of five district typologies have been investigated, namely a reference case and four additional models departing from the base geometry in terms of building height and density (Fig. A.3). The global aspect ratio $H^* = \bar{H}/L_D$ is used to evaluate the average rise of building height $\bar{H}$ in the district, with here $\bar{H} = H$ for cuboids of same heights. The total site coverage $\kappa$ (ratio of the cumulative building footprints to the effective site area) measures the proportion of built to exposed land. The reference configuration (MR18_Mid410) has been designed to depict a semi-compact mid-rise typology ($\kappa = 0.410$; $H^* = 0.18$), according to the Local Climate Zones (LCZ) defined by Stewart and Oke [39]. Two additional districts, MR18_Op292 and MR18_Cp518, having the same global aspect ratios ($H^* = 0.18$, mid-rise) but with a sparser ($\kappa = 0.292$, open) and more abundant ($\kappa = 0.518$, compact) building presence, respectively, have been defined. Another final pair of configurations, labelled LR6_Mid410 and HR36_Mid410, have been modelled, with now a constant total site coverage similar to MR18_Mid410 ($\kappa = 0.410$, semi-compact), but with respective lower ($H^* = \frac{0.18}{3} = 0.06$, low-rise) and higher ($H^* = 2 \times 0.18 = 0.36$, high-rise) average building heights. The restriction of the site to a fixed $L_D \times L_D$ area across all five configurations causes peripheral buildings to have a trimmed effective footprint for some values of $H^*$ and $\kappa$. Any other non-dependent morphological parameters have been kept constant across all five models. The values for main relevant typological and urban design indicators relative to each configuration are detailed in Table A.1. All geometries have been designed using the modelling engine Rhino3D prior to the ray tracing procedure.

Although common features of modern urban landscapes, no semi-transparent (e.g. vegetation) or highly specular (e.g. glazings) materials have been considered here. The
appreciation of their yet clear interplay with sunlight *via* changes in its reflected, transmitted, absorbed or diffused components lies beyond the scope of the present work. Both the directional and spectral dependencies of the model geometry on received irradiance is thus omitted, with all surfaces being supposed grey, opaque and Lambertian. Their radiative properties are in this specific case entirely characterised by a diffuse reflectance, being defined here *via* a spectrally constant Lambertian BRDF<fnref n="1" /> in dedicated files. An asphalt pavement and dark concrete are assumed for the ground (including the buffer band) and building faces materials, respectively. Their corresponding diffuse reflectance have been set accordingly to average values observed for such surfaces in typical urban environments [36], and are respectively equal to $\rho_{gr} = 0.10$ and $\rho_{bl} = 0.20$. No specific temperature is assumed for the considered surfaces as only the shortwave radiation is modelled here (no infrared radiative fluxes). As a result, boundary conditions of each district configuration with the atmospheric layer are entirely defined by their geometries and surface diffuse reflectances.

The same atmospheric conditions as used in Caliot et al. [6] and Le Gall et al. [28] are considered here. To focus on the sole impact of the geometry, a clear sky is assumed, with liquid water contents suspended in the atmosphere being omitted from the calculation to ignore the 3-D cloud cover. The physical properties of the atmospheric layer are indeed entirely described by vertical profiles of temperature, pressure, water vapour mixing ratio, and a combination of other gas constituents ($CO_2$, $CH_4$, $N_2O$, CFC1, CFC2, $O_2$ and $O_3$) [43] using the ecRad radiation scheme for the European Centre for Medium-Range Weather Forecasts (ECMWF) [20]. Implementation of the Rapid Radiative Transfer Model for Global Circulation Models (RRTM-G) [23] in ecRad then allows to compute the spectral optical properties of aerosol particles and *k*-distributions of atmospheric gases over a discrete number of points for a specific set of horizontal layers in the atmosphere from 1-D profiles of characteristic variables [19]. These values are subsequently interpolated to evaluate the radiative properties data, namely the absorption (visible and infrared) and scattering (visible) coefficients, at any local point in the 3-D atmosphere. The location of Paris, France has been selected to depict a mid-high latitude temperate climate.

<footnote n="1">¹Bi-directional Reflectance Distribution Function</footnote>
## 2.2. Simulation of the annual shortwave radiation

The thinness of adopted samplings together with the span of spatial and temporal integration domains especially complicates the collection of experimental irradiance data. No such public database exclusively based on empirical measurements, with similar resolutions and spans, is neither currently available. Propagation of radiation within the different urban configurations are thus simulated here using the free software `htrdr-urban` [5], which is based on an open-source library of independent programs recently proposed for the simulation of radiative transfers in a three-dimensional participating medium [43]. The package has been validated for an infinite idealised urban canyon model against an established radiation scheme based on the radiosity method [6], and employed by the authors in a previous study [28]. `htrdr-urban` adopts a stochastic approach, namely backward Monte Carlo ray tracing, to solve the integral solutions of the three-dimensional Radiative Transfer Equation (RTE). Solar spectral irradiance (SSI) values are provided by the U.S. National Oceanic Atmospheric Administration (NOAA) [7]. Monte Carlo methods (MCM) have been extensively used and validated for the resolution of radiative transfer problems [22, 41] or for the specific modelling of solar radiation [12]. The Monte Carlo procedure implemented in `htrdr-urban` uses the technique of null-collision algorithms (NCA) combined with a tree data structure of atmospheric properties, which makes the calculation time insensitive to the complexity and size of the geometry. Thus, the radiative flux density of surfaces are computed, accounting for their radiative properties (directional and spectral), the multiple scattering in the atmosphere and various reflections [6]. Such a high degree of accuracy is of particular relevancy here, as it lessens the possible gap between simulated data and actual equivalent empirical measurements by limiting the risk for the modelling to ignore a certain extent of the real-world processes involved on the urban scale (e.g. multi-reflections), which could otherwise restrict or mislead the subsequent analysis. By first setting up and validating the proposed approach on a simplified setup, the authors ensure its fair applicability on more complex and realistic urban models in the forthcoming work.

For each of the five district configurations (Section 2.1), the irradiance incident on the four vertical façades and roof of the central building is evaluated (green cuboids in Figs. A.2 and A.3). Values are estimated at each point of a regular evaluation grid located on top of each surface. The grid faces outward and its dimensions match with the total size of the considered surface. The spatial sampling is set to $\Delta x = 1$ m
to appreciate the range of possible variations in the spatial distribution of radiative flux caused by surrounding infrastructures ($N_x \in [108; 648]$ grid points per simulation, depending on the value of $H$). Similar resolutions can be found in previous works with comparable models [15, 28]. The surface field of received radiation is evaluated every 15 minutes ($\Delta t = 15$ min) over a complete typical non-leap year, namely here from 1st January 2019, 00h00 to 31st December 2019, 23h45 ($N_t = 35,040$ timesteps per simulation). Times and dates are specified to htrdr-urban in terms of solar azimuth and elevation angles obtained from the implementation of the NREL Solar Position Algorithm (SPA) [37]. SPA values are given with a $\pm 0.0003^\circ$ uncertainty and evaluated from the site geographic coordinates (48°51'0" N; 2°20'24" E; 49 m alt.) and its weather averages specificities at the ground level ($P = 1013$ Pa; $T = 294.2$ K). Note that average ground pressure and temperature are similar to values set for the atmospheric profiles as generated by ecRad (Section 2.1). Each ray tracing procedure is numerically independent from other estimations at distinct grid points and timesteps, and is set with a resolution of 10,000 Samples Per Pixels (SPP). The simulation is restricted to shortwave radiation, specifically to the spectral band [0.3 µm; 3.0 µm], so that about 96% of the total extra-terrestrial solar radiation is accounted for.

## 2.3. Principal Component Analysis

The existing literature about Principal Component Analysis and analog methods being substantial, the present section focuses on exposing its general principles and mathematical formalism relevant to the context of this research. The authors further refer the reader to Berkooz et al. [2] and Jolliffe [26] for a comprehensive introduction to the method and its applications.

Let $e(\mathbf{x}_j, t_i)$ be the solar radiative flux density [W.m$^{-2}$] received by a point $M_j$ of coordinates $\mathbf{x}_j = (x_j, y_j, z_j)$ from a bounded region in space $\Omega$ at a given time $t_i \in T$, with $T$ a finite period $(i, j \in \mathbb{N}^*)$. $\Omega$ and $T$ are both sampled subspaces of $\mathbb{R}^3$ and $\mathbb{R}$, respectively. The time series of observations at $M_j$ is noted $e_j(t) = \{e(\mathbf{x}_j, t)\}_{t \in T}$ and sensibly assumed to be stochastic. All realisations of $e_j(t)$ in $\Omega$ are considered here, i.e. $\{e(\mathbf{x}, t)\}_{t \in T} = \{e_j(t) : j \in \mathbb{N}^*\}_{t \in T}$, which simply accounts for the overall vector field of irradiance incident on $\Omega$ over the period $T$ (multi-dimensional signal in space). $\{e(\mathbf{x}, t)\}_{t \in T}$ thus constitutes a discrete second-order stochastic process for $t \in T$ defined on the Hilbert space $L_2(\Omega, T)$, with $E[\cdot] = \langle \cdot \rangle_T$ and $\langle \cdot, \cdot \rangle_\Omega = \| \cdot \|^2_\Omega$
its associated temporal average (assumed statistical stationarity and ergodicity) and spatial inner product, respectively.

Principal Component Analysis relies on the Karhunen-Loève theorem stating that the space-time field $e(\mathbf{x}, t)$ can be expanded in an orthonormal basis of deterministic spatial functions $v_k(\mathbf{x})$ spanning $L_2(\Omega)$ (spatial modes) modulated by uncorrelated stochastic time sequences $z_k(t) \in L_2(T)$, called principal components:

$$e(\mathbf{x}, t) = \sum_{k=1}^{N} z_k(t)v_k(\mathbf{x}), \qquad (1)$$

with $N \le \min(N_{\mathbf{x}}, N_t)$ the rank of $e(\mathbf{x}, t)$. Couples $\{(v_k, z_k)\}_{k=1}^N$ are specifically sought for the subspace $\{v(\mathbf{x})\}_{k=1}^N$ to constitute the optimal basis minimising the total mean squared error ($L_2(\Omega, T)$ sense) between the original process and its truncation at any rank $n$:

$$\mathbb{E}\left[\left\|e(\mathbf{x}, t) - \sum_{k=1}^{n} z_k(t)v_k(\mathbf{x})\right\|_{\Omega}^{2}\right] \to 0, \qquad (2)$$

where $n \le N$. Functions $v_k(\mathbf{x})$ are subject to an orthonormality constraint, so that elements in $\{(v_k, \lambda_k)\}_{k=1}^N$ become solutions of the Fredholm eigenproblem:

$$\int_{\Omega} K_{\mathbf{e}}(\mathbf{x}, \mathbf{x}')v_k(\mathbf{x}') d\mathbf{x}' = \lambda_k v_k(\mathbf{x}) \iff (\mathcal{K}_{\mathbf{e}}v_k)(\mathbf{x}) = \lambda_k v_k(\mathbf{x}), \quad k \in \{1, 2, \dots, N\}, \quad (3)$$

where $K_{\mathbf{e}}: \Omega \times \Omega \to \mathbb{R}$ denotes the spatial covariance function of $e(\mathbf{x}, t)$ (definite positive kernel) and $\mathcal{K}_{\mathbf{e}}$ its associated Hilbert-Schmidt integral operator in $L_2(\Omega)$. Principal components $z_k(t)$ are then obtained by expanding the original space-time field $e(\mathbf{x}, t)$ onto their corresponding spatial modes $v_k(\mathbf{x})$:

$$z_k(t) = \langle v_k(\mathbf{x}), e(\mathbf{x}, t) \rangle_{\Omega}, \quad k \in \{1, 2, \dots, N\}, \qquad (4)$$

with their respective contribution to the $L_2(\Omega)$ norm of $e(\mathbf{x}, t)$ (i.e. variance) being directly given by eigenvalues $\lambda_k$. The orthogonality property between elements in $\{v_k(\mathbf{x})\}_{k=1}^N$ especially implies that each amplitude $z_k(t)$ depends exclusively on its corresponding spatial pattern $v_k(\mathbf{x})$ of same rank $k$. It may still be convenient at this point to introduce the expansion coefficients $a_k(t) = \lambda_k^{-1/2} z_k(t)$, which are nothing more than scaled versions of the principal components with now a unit variance.
Equation (1) can thus be rewritten to display the previously hidden eigenvalues $\lambda_k$:

$$ \mathbf{e}(\mathbf{x}, t) = \sum_{k=1}^{N} \sqrt{\lambda_k} a_k(t) \mathbf{v}_k(\mathbf{x}). \quad (5) $$

From a physics aspect, the covariance of variations in the original irradiance field $\mathbf{e}(\mathbf{x}, t)$ is indeed decomposed to identify a new unique set of uncorrelated components $\{a_k(t)\}_{k=1}^N$, being linear combinations of initial time series $\{e_j(t)\}_{j=1}^{N_x}$, but now explaining at best the variability in $\mathbf{e}(\mathbf{x}, t)$ as measured by eigenvalues $\lambda_k$. Their corresponding spatial patterns form a complete orthonormal basis $\{\mathbf{v}_k(\mathbf{x})\}_{k=1}^N$ of $L_2(\Omega)$ pointing towards these main directions of variation, i.e. capturing the most information about space-time dynamics in $\mathbf{e}(\mathbf{x}, t)$ [26]. Characterisation of variations in space and time of the incident irradiance being the very focal point of this research, PCA turns out to be a particularly relevant method here. Note that, by convention, eigenpairs $\{(\mathbf{v}_k(\mathbf{x}), \lambda_k)\}_{k=1}^N$ are sorted by decreasing eigenvalues, i.e. by decreasing order of influence on the global variations. As a results, any truncation of the original irradiance field to a given rank $n \le N$ would eventually contain at least as much information about its variability, i.e. variance, as any other possible linear combinations of orthonormal structures $\mathbf{v}(\mathbf{x})$ from $L_2(\Omega)$. Its reconstruction naturally becomes exact for $n = N$.

The cumulative explained variance $EV^n$ is finally introduced as a measure of the amount of information about the signal's global behaviour contained in its first $n$ empirical modes $\mathbf{v}_k(\mathbf{x})$:

$$ EV^n = \frac{\sum_{k=1}^{n} \lambda_k}{\sum_{k=1}^{N} \lambda_k}, \quad (6) $$

where $\sum_{k=1}^N \lambda_k$ denotes the cumulative variance associated to modes $\mathbf{v}_k(\mathbf{x})$ required to completely describe the original process $\mathbf{e}(\mathbf{x}, t)$. Similarly, the variance explained by a single mode of rank $k$ equals:

$$ EV_k = \frac{\lambda_k}{\sum_{l=1}^{N} \lambda_l}, \quad (7) $$

so that, $EV^n = \sum_{k=1}^n EV_k$.
### 3. Characterising the urban irradiance variability from its dominant eigenmodes

#### 3.1. Global field of annual irradiance

The annual solar radiative flux density incident on all five surfaces of the central cuboid for the reference case MR18_Mid410 is first shown in Fig. A.4. The spatial distribution of annual cumulative irradiance (typical year 2019) over the building facets (Fig. A.4.a) naturally reflects its orientation, with maximal values being observed on the unobstructed roof visible to the entire sky dome. Highest and lowest surface-averaged cumulative irradiance are obtained on the southern and northern walls, respectively, while equivalent distributions are shown for the eastern and western faces.

The temporal evolution of the space-averaged irradiance received by the entire building (Fig. A.4.b) is further coherent with the intrinsic daily and seasonal sun path for the current location (Paris, France). Seasons (horizontal axis) are clearly depicted, with the overall incident solar flux density peaking at the summer solstice and symmetrically decreasing towards the equinox to reach its minimal value around the winter solstice. Similarly, the building hourly-averaged irradiance (vertical axis) describes a bell-shaped distribution centred around midday across the entire annual period, with still a light drop being observed at noon. This specific variation may be explained by the roof becoming the only apparent facet receiving direct solar radiation at this specific period of the day, with almost no radiative flux incident on other vertical façades.

Plots of the daily irradiance amplitude at specific points of the cuboid faces for critical dates, i.e. solstices and equinoxes, (Fig. A.4.c) give additional insights about the building orientation. The foremost part of the daily radiative flux arriving on the east-facing wall (magenta area) is collected in the morning. Likewise, its incidence on the southern façade (green area) instead spreads throughout the entire day for the main part of the year, with maximal values being reached around midday where the sun directly faces the façade. Note here the lower daily irradiance received at the summer solstice (21st June) than on 21st March, caused by overly high sun elevation angles observed over this period shrinking the contribution of direct solar radiation.

The influence of the surrounding geometry on the spatiotemporal distributions is eventually highlighted in the results, with a drop in the annual cumulative irradiance on all vertical facets of the cuboid towards the bottom of the façades (Fig. A.4.a). This spatial variations are especially emphasised for the most exposed walls, but only
slightly appear on the northern face mainly receiving diffuse light. Note also the abrupt variation of irradiance over the green patch on the south-facing façade on 22nd December appearing around 10:00 am (Fig. A.4.c), showing a sharp cut in the received radiation. This is much likely caused by the presence of the building immediately facing the central cuboid on its southern side, which blocks solar rays whose incident angles are low over the main part of the day in winter for the chosen location.

The Discrete Fourier Transform (DFT) of the global irradiation field received by the entire envelope further highlights its characteristic frequencies in the temporal domain (Fig. A.5). Harmonics associated to daily and sub-daily periods become clearly visible in the signal spectrum, suggesting the prevalence of these frequencies contribution on its overall behaviour in time.

## 3.2. Preprocessing of the simulated irradiance field

It is very often the case in some disciplines, where the dynamics of multi-dimensional phenomena is of interest, that the stochastic process is centred with respect to its time average prior to the orthogonal decomposition (e.g. atmospheric science [17], fluid dynamics [9]). Translating the initial values to have null temporal means especially ensures that PCA examines the departure of the input signal from its average behaviour over the analysis period, with the first principal component actually reflecting its maximum variance rather than its average value. Note however that the centering procedure is not mandatory but rather depends on the very nature of the investigated dataset. Here, the time average of the signal may still contain useful information. Because the simulation is performed over an entire year with a sub-daily resolution, centering the temporal evolution of irradiance received at each point of the measurement grid around their yearly average value indeed claims little relevancy. The periodic nighttime patterns, during which the amount of incident solar radiation naturally drops to nought, indeed appear as gaps of useful knowledge as they do not provide any additional information about the impact of the urban fabric on variations of the local solar resource. Instead, because of the deterministic aspect of sunlight variations, presenting daily and seasonal cycles, the irradiance received over the entire period T by each grid point in the absence of any surrounding geometry and under clear sky conditions has been subtracted to the original simulated values. The (i, j)-th element of the initial dataset $e(x, t)$ can therefore be decomposed as follow:

$$e(x_j, t_i) = \tilde{e}(x_j, t_i) - e_0(x_j, t_i), \quad i \in \{1, 2, \dots, N_t\}, \quad j \in \{1, 2, \dots, N_x\}, \quad (8)$$
where $\tilde{e}(\mathbf{x}_j, t_i)$ and $e_0(\mathbf{x}_j, t_i)$ denote the simulated irradiance at point $M_j(\mathbf{x}_j)$ for the timestep $t_i$ with and without geometrical obstacles, respectively. Hence, the annual field of irradiance detailed in Section 3.1 refers to elements $\tilde{e}(\mathbf{x}_j, t_i)$ in Eq. (8).

### 3.3. First eigenmodes of variability

Decompositions of the irradiance field received by the central cuboid in all five urban district configurations (see Section 2.1) are now exposed. Namely, the extracted PCA modes $\mathbf{v}_k(\mathbf{x})$ and their related expansion coefficients $a_k(t)$ are interpreted as regards spatiotemporal variations in $e(\mathbf{x}, t)$. Spatial modes have been reconstructed and normalised in the range [-1; 1] to display variations in the surface irradiance on the different facets of the building. Similarly, time series $a_k(t)$ have been rearranged to express their respective daily magnitude (24 hours) as a function of the day in the year (365 days) (Figs. A.7 to A.11). A diverging colormap is used for the sake of visualisation and results are interpreted as absolute evolutions (unsigned variations). The analysis is restricted to the first six modal ranks $k \in \{1, 2, \dots, 6\}$, which have been found to capture most of the information about the signal variability (see Fig. A.6) and deemed sufficient here for a sound validation of the approach. While higher-order modes (i.e. $k \ge 7$) may still provide useful insights about the interaction of the urban environment with the incident radiation, it would likely apply in the presence of more real-like urban models displaying a higher degree of structural and radiative intricacy, thus widening the range of variability factors.

#### 3.3.1. Separate description of space and time variables on characteristic scales

A first essential outcome inherent to modal decomposition is the separate representation of spatial and temporal variations, which is clearly visible from the respective distributions of $\mathbf{v}_k(\mathbf{x})$ and $a_k(t)$ in Figs. A.7 to A.11. Each expansion coefficient $a_k(t)$, and equivalently principal component $z_k(t)$, of rank $k$ relates to a characteristic time scale for the corresponding spatial mode $\mathbf{v}_k(\mathbf{x})$. Overall, an expected correlation is found between the order of decomposition, i.e. magnitude of $k$, and the frequency band depicted. First eigencoefficients $a_k(t)$, associated with greatest variances, usually portray the signal variability at low frequencies while components with lower ranks describe finer variations. Temporal frequencies described by the dominant eigenmodes of the global irradiance field in fact seem to fit the principal harmonics populating its DFT spectrum in Fig. A.5. This suggests some sort of degeneracy for these modes, betraying the temporal periodicity of the signal. Note that this observation is not
trivial, as there is no mathematical requirement for each rank-k mode of an arbitrary
process to be associated to a single temporal frequency.

In the present work, the dominant mode is commonly associated to annual vari-
ations of the received solar radiative flux, where the impact of seasonal cycles dis-
tinctly appears, e.g. $a_1(t)$ for **MR18_Mid410** (Fig. A.7.a), **MR18_Cp518** (Fig. A.8.b),
**HR36_Mid410** (Fig. A.10.b). Corresponding spatial distributions $v_1(x)$ naturally de-
pict a decreasing homogeneous ground-to-roof evolution of the irradiance received by
the building, with an expected peak magnitude at the bottom of vertical façades where
the direct light exposure is the weakest, or equivalently, the influence of immediate sur-
rounding obstructions is the most significant. Note here the higher gradients observed
on the most exposed southern facets.

Subsequent ranks target higher frequencies relating to daily variabilities. The
morning/afternoon sharing of irradiance is especially visible for $k=2$ in the same three
configurations ($a_2(t)$ — Fig. A.7.b, Fig. A.8.d, Fig. A.10.d), where spatial modes $v_2(x)$
now relate exclusively to the progressive daily switch in received radiation between the
east- and west-facing walls. Midday is further highlighted by the third modal rank
($a_3(t)$ — Fig. A.7.c, Fig. A.8.f, Fig. A.10.f), with the spatial variability patterns $v_3(x)$
additionally distinguishing between the southern façade and eastern and western faces.

This hierarchy is yet not rigid and low frequency ranges may still be described
at higher-order ranks. This is especially the case for configurations **MR18_Op292** and
**LR6_Mid410**, where the first two expansion coefficients have been switched with respect
to aforementioned models. The principal mode is now associated to a daily variability
of solar irradiance ($a_1(t)$ in Fig. A.8.a and Fig. A.10.a), while second-order spatial
variations have a magnitude evolving annually ($a_2(t)$ in Fig. A.8.c and Fig. A.10.c).
In other words, the daily changes in the solar resource between the morning and the
afternoon prevails here in the spatial variation of irradiance on the central cuboid.
This further informs about the predominance of their corresponding spatial modes in
the overall variability. This observation is not surprising as the cumulative annual
irradiance received at each measurement point in an equivalent building-free environ-
ment has been subtracted to the original signal prior to PCA (see Section 3.2). The
presence of the annual sun path in variations of incident irradiance is therefore further
erased for configurations **MR18_Op292** and **LR6_Mid410**, where the evaluated façades
are very much visible to the sky dome, than for any other typologies in which im-
mediate geometries provide higher degrees of obstruction. Note however that in all
configurations but the high-rise model HR36_Mid410, the variances explained by the
first two modes differ only slightly (< 5%), so does the weight of their impact on the
spatial distribution of irradiance.

3.3.2. Impact of the district morphology

For all configurations, distributions depicted by the first three spatial modes **v**<sub>1-3</sub>(**x**)
remain relatively uniform, with faint homogeneous vertical variations of the radiative
solar flux across all façades. No real distinction is made here between individual build-
ings. Surrounding infrastructures are instead considered as a single global obstacle
for incident radiation. Changes in its reflected and diffuse components are mainly
portrayed by these first modes. Morphological specificities of each district yet remain
palpable, with slight differences being induced on certain modes of same rank **v**<sub>k</sub>(**x**)
by a modification of the total site coverage κ or global aspect ratio *H*\* (e.g. **v**<sub>2</sub>(**x**) in
Fig. A.8 and Fig. A.10). Rapid spatial variations of irradiance caused by the shadow-
ing effect of neighbouring structures is however weakly apparent in dominant patterns
**v**<sub>1-3</sub>(**x**).

From *k* = 4, non-homogeneous variations in the direct radiative component start
to be clearly represented by PCA modes (Fig. A.6), as shown by visualisations of
**v**<sub>4-6</sub>(**x**) in Fig. A.7.d-f, A.9 and A.11. For all five configurations, the same spatial
patterns are encountered across all of their three ranks *k* = 4, 5 and 6 for the eastern
and western walls of the central cuboid, now picturing bi-directional surface distribu-
tions. The presence of bordering cuboids on both its east and west sides is depicted
by their footprints on the central building, e.g. **v**<sub>4-6</sub>(**x**) in Fig. A.7.d-f. The obvious
symmetry of observed patterns further highlights the homogeneity of district arrange-
ments. In addition, an imprint is created by the southern geometry to the central
building on its south façade. This is again portrayed by specific modes, e.g. **v**<sub>4,6</sub>(**x**) for
**MR18_Mid410** (Fig. A.7.d,f.) and **LR6_Mid410** (Fig. A.11.a,e.), **v**<sub>4,5</sub>(**x**) for **MR18_Cp518**
(Fig. A.9.b,d.) and **HR36_Mid410** (Fig. A.11.b,d.), suggesting the obstruction of sun
rays with low angles of incidence. Other patterns observed on the south facet across the
different configurations, e.g. **v**<sub>5</sub>(**x**) for **MR18_Mid410** (Fig. A.7.e), **v**<sub>6</sub>(**x**) for **MR18_Op292**
(Fig. A.9.e), **MR18_Cp518** (Fig. A.9.f) and **HR36_Mid410** (Fig. A.11.f), further intimate
the presence of obstacles. These are caused by the alternate blockage of sun rays
by the regular arrangement of southern obstacles. The presence of streets on either
sides of the district infrastructures is particularly evident through these modes. It is
worth noting that these southern patterns are repeatedly found across the evaluated
configurations, but at different decomposition ranks *k*, suggesting various degrees of
influence on the received irradiance variability.

The change in the total site coverage $\kappa$ is also plainly apparent in the distributions
of same-rank spatial modes $v_k(\mathbf{x})$. A decrease in $\kappa$, i.e. in the building-to-building
distance, progressively draws the footprint of closest obstructions on relevant façades,
as shown by the evolution of $v_4(\mathbf{x})$ on the eastern wall from MR18_Cp518 (Fig. A.9.b) to
MR18_Op292 (Fig. A.9.a). Similarly, an increase in the global aspect ratio $H^*$ generally
shifts the variability pattern towards the top of the façade, subject to higher direct
sunlight exposure than areas close to the ground (e.g. $v_5(\mathbf{x})$ on the east-facing wall in
Fig. A.11.c, Fig. A.7.e, and Fig. A.11.d).

Expansion coefficients $a_{4-6}(t)$ associated to the last three spatial modes $v_4(\mathbf{x})$, $v_5(\mathbf{x})$ and $v_6(\mathbf{x})$ interestingly hold similar evolutions than for the first three dominant ranks in most cases. This parallel is of some importance here as it further stresses the distinction between the homogeneous and non-homogeneous states of variability induced by the district geometries (Fig. A.6). Non-homogeneous changes, which are portrayed by eigenmodes at ranks $k = 4, 5$ and $6$ are indeed represented on the same frequency ranges as previously tackled by homogeneous variations (ranks $k = 1, 2, 3$). Note that absolute magnitudes of time series $a_{4-6}(t)$ are still important over those of coefficients $a_{1-3}(t)$ (disregarding the variance associated to each mode), and some distributions have been switched between specific ranks. Looking at the compact mid-rise district MR18_Cp518, the amplitude of daily variations observed at both equinoxes have doubled from $a_1(t)$ (Fig. A.8.b) to $a_4(t)$ (Fig. A.9.b). An equivalent note can be made for subsequent ranks of the same model, with the distributions depicted in $a_2(t)$ and $a_3(t)$ (Fig. A.8.d,f) being encountered again at higher orders but in $a_6(t)$ and $a_5(t)$, respectively (Fig. A.9.d,f).

Results obtained for the open mid-rise typology MR18_Op292 yet slightly depart
from the aforementioned comments. Spatial variations at ranks $k \ge 4$ indeed appear
to occur on finer time scales reaching hourly resolutions (see $a_{4-6}(t)$ in Fig. A.9.a,c,e).
Neighbouring obstructions having here a limited influence on the solar radiative flux
received by the central building compared to other configurations with higher build-
ing aspect ratios (see Table A.1), changes in the direct radiation component remain
weak and unnoticed at low frequencies explaining seasonal or global daily variations.
Non-homogeneous spatial variations are indeed brought up for this arrangement by
modes on bi-hourly-resolving scales or finer. This might further explain the already
high magnitudes of the first three expansion coefficients $a_{1-3}(t)$ (Fig. A.8.a,c,e) with
respect to the denser mid-rise models **MR18_Mid410** and **MR18_Cp518**. Note for in-
stance the rank-four temporal distribution $a_4(t)$ observed for **MR18_Cp518** (Fig. A.9.b)
now describing the evolution of $v_2(\mathbf{x})$ in the open mid-rise model (Fig. A.8.c). The
magnitude is also more equally spread across all six expansion coefficients $a_{1-6}(t)$ for
the semi-compact low-rise typology **LR6_Mid410** (Fig. A.10.a,c,e and Fig. A.11.a,c,e),
holding an equivalently low building aspect ratio, than for other configurations. Most
unequal repartitions between ranks 1–3 and 4–6 are indeed observed for the semi-
compact high-rise (**HR36_Mid410**) and compact mid-rise (**MR18_Cp518**) districts, for
which obstructions are likely to have a greater influence on the distribution of the
incident radiation.

Overall, variations on northern façades, being the least exposed to direct sun-
light, are usually negligible across the various models, with some slight variations
still being observed for the most detrimental typologies, e.g. $v_{4,5}(\mathbf{x})$ for **MR18_Cp518**
(Fig. A.9.b,d). Similarly, no variations in the solar flux density on the unobstructed
roof surfaces is sensibly observed in all configurations, regardless of the decomposition
rank $k$.

3.4. *Distribution of the explained variance across modal ranks*

The evolution of eigenvalues $\lambda_k$ and normalised cumulative explained variance $EV^k$ for corresponding eigenmodes $v_k(\mathbf{x})$ have both been plotted as a function $k \in \{1, 2, \dots, 10\}$ for all district typologies (Fig. A.6). Overall, the first six PCA modes explained at least 77.57% (MR18_Op292) of the global signal's behaviour, with the highest ratio being observed for the semi-compact high-rise district HR36_Mid410 (92.69%).

Variances explained by the three dominant eigenmodes ($k \le 3$) remain globally sub-
stantial over values obtained at higher decomposition ranks where non-homogeneous
variations of the incident radiation caused by surrounding geometries start to be ac-
counted for, as shown by the drop in the evolution of $\lambda_k$ (log-scale) between $k = 3$ and
$k = 4$ for all cases (solid lines in Fig. A.6). In other words, the direct contribution of
individual urban geometries to changes in the global field of irradiance on the central
building stays weak over homogeneous variations in its diffuse and reflected compo-
ents induced by the overall district itself. This could have been expected given the
homogeneity of the district arrangements and the simplicity of input parameters (e.g.
materials optical properties, climate conditions) considered in the present study.
Variances $\lambda_k$ are intrinsically proportional to the total building envelope, i.e. to the overall received solar radiative flux (solid lines in Fig. A.6). This is relatively well highlighted by LEV spectra for ranks $k \ge 4$, with configurations of same global aspect ratios $H^*$ (round markers) holding equivalent magnitudes. For lower-order modes, departures are still observed between the different typologies. Namely, values of $\lambda_k$, or equivalently $\log(\lambda_k)$, tend to be higher in average for configurations where surrounding geometries are likely to have a bigger impact on the obstruction of the incident shortwave radiation, i.e. MR18_Cp518 and HR36_Mid410, than in open and/or low-rise typologies in which the sky dome remains more visible to the central building façades (MR18_Op292 and LR6_Mid410).

Similarly, the evolution of the model’s cumulative explained variance is sharper for packed and high geometries (dashed lines on Fig. A.6), with the drop at the characteristic rank $k = 4$ increasing proportionally to the degree of obstruction on the central cuboid. For obstruction-favourable districts (i.e. MR18_Cp518, HR36_Mid410), a more important part of the global variance is indeed explained by dominant modes related to quasi-uniform surface flux distributions (i.e. $k \le 3$ here), than for sparser typologies (i.e. MR18_Op292, LR6_Mid410) where the signal variance is more equally spread across the first six eigenmodes. In other words, fewer PCA modes are required to explain a given amount of information about the global signal variability in the former than latter configurations, where geometries are less likely to obstruct incident solar rays. This could have been expected as the contribution of the sun course on the variation of received irradiance has been subtracted to the dataset prior to the modal decomposition (see Section 3.2). These observations correlate with evolutions of absolute amplitudes $|a_k(t)|$ in Section 3.3.2.

### 3.5. Limitations and perspectives

Overall, all five evaluated districts have held here a perfect structural homogeneity, with all materials being Lambertian and entirely defined by their constant diffuse reflectance. It is yet reasonably supposed that in the presence of real-like heterogeneous district configurations, seen as medleys of various elaborate shapes with diverse non-trivial materials (e.g. glazings, vegetation), first modal ranks associated to highest variances would be appropriately populated by eigenmodes now relating to prevalent urban variables. Thus getting the upper hand over current dominant homogeneous variations in the diffuse and reflected irradiance components induced by the global district structure. A higher number of modes would also likely be required to capture
an equivalent amount of information about the signal’s global variations, more than three-quarter of its behaviour being already explained here by the first six PCA components in the worst case open mid-rise configuration (EV⁶ = 77.57%). Naturally, an increase in the model complexity would have intrinsically made eigendecomposition results harder to analyse, leading to more complex spatial patterns and a possibly higher range of effective temporal frequency bands associated to a wide set of variables. Because PCA fundamentally captures structural averages and not localities, the distinction between individual variables might become specifically difficult.

Driving away from controlled homogeneous typologies holding a limited range of possible variables was yet out of the scope of this preliminary study, whose ultimate objective restricts to the first robust assessment and validation of the approach. Findings obtained here still soundly demonstrate the potential and efficiency of variance-based orthogonal decompositions like PCA to analyse the variations of solar radiation in the complex urban environment at high spatiotemporal resolutions. Spatial areas subject to high degrees of variability and related most impactful variables on characteristic temporal frequencies are identified. First extracted information paves the way for an interpretation of eigenmodes in a richer urban context, where a plethora of elements may benefit the variability of solar radiative flux density. No reliable analysis is yet possible here without an additional comprehensive validation of the approach on more realistic urban models granted local heterogeneity in their morphological parameters and surface radiative properties.

Future work would thus involve the validation of the present approach and first observations over more complex district configurations, holding structural and material properties closer to real urban environments. Namely, the introduction of some heterogeneity across main urban typological characteristics (e.g. buildings arrangement, orientation, density, dimensions) would add on obvious disruption to the incident field of shortwave radiation, and is thus expected to distort extracted eigenmodes. Common urban materials such as highly-specular glass panes or pervious vegetated elements are also likely to introduce noticeable changes here. Evaluating the impact of this geometrical and material heterogeneity would allow for a sound appreciation of prevalent variables, building on current observations. An extension of the decomposition to specific groups of surfaces or district areas may help to this end, while expanding the variability analysis beyond the building envelope. Performing a district-wise decomposition
would be especially helpful for the identification of intra-urban regions subject to high
spatial and temporal variations of irradiance, and for the subsequent characterisation
of this variability. The additional high heat storage capacity held by some materials
(e.g. building façades, road tarmac) further contributes to increasing the amount of
terrestrial infrared radiation exchanged with the atmosphere (infrared radiation trap-
ping) [6, 3]. This participates to the so-called Urban Heat Island (UHI) effect, when
coupled to the overall structural shading and other urban-specific disturbances (e.g.
perturbation of the local wind pattern), establishes a specific local climate [25, 4]
adding on the variability of the solar resource. The extension of the proposed meth-
odology to longwave radiation may thus be explored in this scope. Overall, increasing
the model complexity would necessitate an investigation to higher-order decomposi-
tion levels than the current sixth rank. A specific hurdle would be here to appreciate
the sufficient number of modes being worth considering for the correct analysis of the
signal variability.

The nature of the modal decomposition algorithm can be further explored. As high-
lighted in the introduction (Section 1), a plethora of dimensionality reduction methods
exist and are still undergoing active research in various disciplines [17, 18]. PCA un-
derperforms in representing local phenomena, as stated hereinabove, and may rapidly
become limited regarding the interpretation of the physics behind extracted modes
when dealing with intricate phenomena of high dimensionality [26]. Other decompos-
ition algorithms may thus appear more relevant in the present context by alleviating
some weaknesses of PCA. These are worth being implemented for their relevancy
to be assessed in terms of results interpretation over the current method. Specifically,
the consideration of a *priori* known spatiotemporal characteristics of the input data-
set, such as temporal periodicities (e.g. daily and seasonal cycles), *via* an extension to
higher-order decompositions is a possible line of investigation. The nature of individual
modes, and so their underlying physics, are directly dependent on the decomposition
algorithm. Choosing the appropriate method therefore remains fundamental here.

4. Conclusions

This paper has presented an original approach for the characterisation of solar
radiation variability in an urban context based on modal decomposition. A so-called
Principal Component Analysis (PCA) has been applied to the annual shortwave radi-
ative flux density modelled on the envelope of a cuboid building located at the centre
of a theoretical district, with a square metre spatial resolution and sub-hour temporal
sampling. Datasets have been processed to restrict the impact of natural variations
of irradiance prior to the decomposition. To validate the proposed methodology, four
homogeneous urban typologies with diverging global aspect ratios (average buildings
height) and total site coverage (buildings density) have been investigated against a
fifth reference case.

For the first time, results strictly demonstrate the applicability of orthogonal decomposition methods for apprehending the separate variability of the urban solar resource in both space and time, and identifying its main factors of variation. While the investigation is performed on the building envelope, its district-wise extension can be reasonably considered. Main conclusions are listed hereinafter:

* PCA intrinsically allows for a separate representation of spatial and temporal variables for the initial multi-dimensional irradiance. Spatial eigenpatterns $v_k(x)$ are successively extracted based on a variance-maximisation criterion and ranked by increasing influence on the global variability, as measured by eigenvalues $\lambda_k$. The projection of the original process onto each mode $v_k(x)$ provides their corresponding expansions coefficients $a_k(t)$. Every rank-k contribution $\sqrt{\lambda_k}a_k(t)v_k(x)$ eventually informs about a specific amount of the global signal's behaviour from isolated pictures of its correlated spatial and temporal evolutions. This specifically assists in identifying areas of the building envelope that experience important spatial and temporal variations of incident solar radiation.

* Respective decomposition ranks $k$ have been found to focus on distinct temporal frequency bands, providing information about seasonal and daily trends in the irradiance variability.

* Important corresponding changes in the local distribution of irradiance are described by spatial modes $v_k(x)$, informing about prevalent obstructions. The apparent surface of immediate peripheral structures visible to the central cuboid resulting from the blocking of incident radiation is pictured in their two-dimensional distributions.

* Information about the global district typology is extracted, with any change in the total site coverage $\kappa$ and the global aspect ratio $H^*$ inducing consistent evolutions of dominant variability patterns on relevant façades. Obstruction-favourable building organisations, namely compact and high-rise aggregates, are
identified. The respective influence of urban morphological specificities as regards to the global irradiance variability in space and time is further suggested by the repeated extraction of identical modes across various façades and/or district configurations. The current homogeneity of buildings arrangements is naturally reflected here by symmetries and complementarities observed between modes on both east-west and north-south axes. In the presence of a more sophisticated heterogeneous district configurations, the latter are expected to pinpoint structural discrepancies with the apparition of some divergences.

* Variations in the diffuse radiative component are clearly distinguished from the blocking of direct radiation by urban geometries through two *states of variability*, as respectively described by dominant (homogeneous variations) and subsequent less-impactful (non-homogeneous variations) eigencomponents.

* Distribution of the variability explained by individual modes further informs about the geometrical intricacy, and distinguishes between disruptions to the diffuse and direct radiative components.

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## Appendix A. Confrontation of *htrdr-urban* against empirical data

The model of radiative transfer *htrdr-urban* has been validated and used in recent research (see Section 2.2). To strengthen its fair adoption, and alleviate the impossibility of collecting experimental data in our context, a simple dedicated validation of the code against empirical irradiance measurements has been additionally performed.
The shortwave downward irradiance (SWD) received by a single point facing upwards has been evaluated with **htrdr-urban** for various days, with a timestep $\Delta t = 1$ min. Results for the 8th, 11th and 22nd July 2023 are displayed here (red plots in Fig. A.12). These are compared to daily distributions of SWD measurements collected by the Baseline Surface Radiation Network (BSRN) [10] (blue curves in Fig. A.12). The BSRN, which currently gathers 58 active monitoring stations worldwide, provides one of the highest quality solar irradiance database. The location of Palaiseau, France (48°42′46.8″ N; 2°12′28.8″ E; 156 m alt.) has been selected. For the sake of comparison, note that sky and sun luminance distributions are not computed here from the NOAA SSI (see Section 2.2), but tabulated values of DNI and DHI in the BSRN dataset have instead been concurrently fed to **htrdr-urban** for each timestep, alongside the respective air temperature and atmospheric pressure at the station level. Further information about the monitoring station (e.g. device, site specificities) can be accessed online [16]. The same computation setup as detailed in Section 2.2 has otherwise been adopted.

The fit between experimental BSRN measurements (blue curves) and corresponding **htrdr-urban** simulations (red curves) is quasi-exact. Periods of clear sky (e.g. 11th July 2023 morning, Fig. A.12c) and cloudy days showing rapid fluctuations of received SWD (e.g. 22nd July 2023, Fig. A.12e) are both accurately described by the modelling engine (accordingly to BSRN weather data). Significant differences between simulated and empirical data repeatedly observed in the early morning (Figs. A.12a, A.12c and A.12e) are caused by the meant restriction of the MCM simulation in **htrdr-urban** to positive sun elevation angles, which is not the case for BSRN data. Small gaps are also observed in **htrdr-urban** spectra (e.g. Fig. A.12b), where DNI and DHI values were missing in the BSRN dataset (if so, set to 0 W.m⁻²). As observed from error bars in Fig. A.12, the overall absolute difference between simulated and collected data remains considerably small. An average absolute error of 5.06 W.m⁻², 10.22 W.m⁻² and 6.06 W.m⁻² is obtained for daily distributions of SWD on the 8th, 11th and 22nd July 2023, respectively. When early morning gaps are not considered, these drop to 2.49 W.m⁻², 3.93 W.m⁻² and 4.50 W.m⁻², which account for about 0.78 %, 1.28 % and 1.62 % of the daily average shortwave downward irradiance, respectively.

Overall, data simulated with **htrdr-urban** proves to be in excellent agreement with benchmark solar irradiance measurements in the real urban fabric from the BSRN database, with an average difference of approximately 1 %.
<figure><img src="image_2.png" /><figcaption>Figure A.1 Illustration of possible radiative events occurring within a typical urban environment.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure A.2 Three-dimensional model of a representative symmetrical homogeneous district. The visualisation is given for the reference case MR18_Mid410. The central building for which the field of incident irradiance is evaluated is highlighted in green. The buffer band is shown in magenta. The district orientation is provided by the north arrow.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure A.3 Variation in the global aspect ratio $H^*$ (LR6, MR18, HR36) and total site coverage $\kappa$ (Op292, Mid410, Cp518) across the input configurations. The central building for which the field of incident irradiance is evaluated is highlighted in green. The buffer band is shown in magenta.</figcaption></figure>
<table><caption>Table A.1 Main typological and urban design parameters for the five district configurations. A district width of $L_D$ = 100 m is assumed.</caption><thead><tr><th></th><th>MR18_Mid410</th><th>MR18_Op292</th><th>MR18_Cp518</th><th>LR6_Mid410</th><th>HR36_Mid410</th></tr></thead><tbody><tr><td>LCZ type [39]</td><td>Semi-<br>compact<br>mid-rise</td><td>Open<br>rise</td><td>mid-<br>Compact<br>mid-rise</td><td>Semi-<br>compact<br>low-rise</td><td>Semi-<br>compact<br>high-rise</td></tr><tr><td>Parameter</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Average building height <i>H̄</i> [m]</td><td>18</td><td>18</td><td>18</td><td>6</td><td>36</td></tr><tr><td>Global aspect ratio <i>H</i>*a [-]</td><td>0.18</td><td>0.18</td><td>0.18</td><td>0.06</td><td>0.36</td></tr><tr><td>Number of stories<sup>b</sup> [-]</td><td>6</td><td>6</td><td>6</td><td>2</td><td>12</td></tr><tr><td>Street width Δ<i>L</i><sub>D</sub> [m]</td><td>9</td><td>23</td><td>7</td><td>9</td><td>9</td></tr><tr><td>Total site coverage κ<sup>c</sup> [-]</td><td>0.410</td><td>0.292</td><td>0.518</td><td>0.410</td><td>0.410</td></tr><tr><td>Floor area ratio<sup>d</sup> [-]</td><td>2.46</td><td>1.75</td><td>3.11</td><td>0.82</td><td>4.92</td></tr><tr><td>Building aspect ratio<sup>e</sup> [-]</td><td>2.00</td><td>0.78</td><td>2.57</td><td>0.67</td><td>4.00</td></tr><tr><td>Shape factor<sup>f</sup> [-]</td><td>0.22</td><td>0.22</td><td>0.22</td><td>0.22</td><td>0.22</td></tr></tbody></table>

<footnote>a Ratio of the average building height $\bar{H}$ to the district length $L_D$.</footnote>

<footnote>b Number of stories per building, including the ground floor. A 3 m floor-to-ceiling distance is assumed.</footnote>

<footnote>c Ratio of the cumulative building footprints to the effective site area.</footnote>

<footnote>d Ratio of the cumulative floor area across all buildings stories to the total site area (without the buffer band).</footnote>

<footnote>e Ratio of a building height $H$ to its distance to neighbouring buildings $\Delta L_D$.</footnote>

<footnote>f Ratio of the cumulative area of a building façades (i.e. envelope) to the building volume.</footnote>
<figure><img src="image_5.png" /></figure>

Figure A.4 Spatial distribution of the cumulative irradiance [kW h.m⁻².yr⁻¹] received by the central cuboid in MR18_Mid410 over the typical year 2019 (a) and corresponding building-averaged annual amplitude [W.m⁻²] (b). The daily building-averaged irradiance evolution on two 1 m² areas of the south- (green) and east-facing (magenta) façades, respectively, is further given for specific dates, namely both solstices (winter — 22nd December; summer — 21st June) and the spring equinox (21st March) (c). The building-averaged irradiance incident on the roof surface for 21st June (light blue) is shown for comparison. Spatial axes in (a) refer to the coordinate system defined in Fig. A.2, with the radial axis matching with the (O, z) direction (unfolded view).
<figure><img src="image_6.png" /><figcaption>Figure A.5 Discrete Fourier Transform (DFT) of the global field of annual irradiance $e(x, t)$. The plotted amplitude $|\hat{e}(x, t)|$ (real value) has been normalised by $N_t/2$ the effective number of samples in the temporal domain.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure A.6 Log-Eigenvalue (LEV) diagram [26] (solid lines, left axis) and normalised cumulative EV spectra (dotted lines, right axis) for the five assessed district typologies. Plots are restricted here to the first $k = 10$ PCA modes.</figcaption></figure>
$$ (a) k = 1 - EV_1 = 33.39\% $$

<figure><img src="image_8.png" /></figure>

$$ (b) k = 2 - EV_2 = 30.87\% $$

<figure><img src="image_9.png" /></figure>

$$ (c) k = 3 - EV_3 = 20.15\% $$

<figure><img src="image_10.png" /></figure>

$$ (d) k = 4 - EV_4 = 2.18\% $$

<figure><img src="image_11.png" /></figure>

$$ (e) k = 5 - EV_5 = 1.51\% $$

<figure><img src="image_12.png" /></figure>

$$ (f) k = 6 - EV_6 = 1.50\% $$

<figure><img src="image_13.png" /></figure>

Figure A.7 First six spatial modes $v_k(x)$ and related expansion coefficients $a_k(t)$ ($k \in \llbracket 1; 6 \rrbracket$) for the reference district MR18_Mid410. Spatial axes refer to the coordinate system defined in Fig. A.2, with the radial axis matching with the $(O, z)$ direction (unfolded view).
<figure><img src="image_14.png" /><figcaption>(c) $\kappa = 0.292$, $k = 2 - EV_2 = 21.75\%$</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(e) $\kappa = 0.292$, $k = 3 - EV_3 = 18.22\%$</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>(b) $\kappa = 0.518$, $k = 1 - EV_1 = 35.65\%$</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(d) $\kappa = 0.518$, $k = 2 - EV_2 = 31.60\%$</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>(f) $\kappa = 0.518$, $k = 3 - EV_3 = 20.13\%$</figcaption></figure>

<figure><img src="image_19.png" /></figure>

Figure A.8 Spatial modes $v_{1-3}(x)$ and related expansion coefficients $a_{1-3}(t)$ for the open (MR18_Op292, $\kappa = 0.292$ — left column) and compact (MR18_Cp518, $\kappa = 0.518$ — right column) mid-rise typologies. Spatial axes refer to the coordinate system defined in Fig. A.2, with the radial axis matching with the $(O, z)$ direction (unfolded view).
<figure><img src="image_20.png" /><figcaption>(c) $\kappa = 0.292$, $k = 5$ - $EV_5 = 4.18\%$</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>(e) $\kappa = 0.292$, $k = 6$ - $EV_6 = 3.98\%$</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>(b) $\kappa = 0.518$, $k = 4$ - $EV_4 = 2.17\%$</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>(d) $\kappa = 0.518$, $k = 5$ - $EV_5 = 1.30\%$</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>(f) $\kappa = 0.518$, $k = 6$ - $EV_6 = 1.25\%$</figcaption></figure>

<figure><img src="image_25.png" /></figure>

Figure A.9 Spatial modes $v_{4-6}(x)$ and related expansion coefficients $a_{4-6}(t)$ for the open (MR18_Op292, $\kappa = 0.292$ — left column) and compact (MR18_Cp518, $\kappa = 0.518$ — right column) mid-rise typologies. Spatial axes refer to the coordinate system defined in Fig. A.2, with the radial axis matching with the $(O, z)$ direction (unfolded view).
<figure><img src="image_26.png" /><figcaption>(c) $H^* = 0.06$, $k = 2$ - $EV_2 = 27.26\%$</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>(e) $H^* = 0.06$, $k = 3$ - $EV_3 = 21.88\%$</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>(b) $H^* = 0.36$, $k = 1$ - $EV_1 = 38.01\%$</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>(d) $H^* = 0.36$, $k = 2$ - $EV_2 = 31.49\%$</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>(f) $H^* = 0.36$, $k = 3$ - $EV_3 = 19.07\%$</figcaption></figure>

<figure><img src="image_31.png" /></figure>

Figure A.10 Spatial modes $v_{1-3}(x)$ and related expansion coefficients $a_{1-3}(t)$ for the semi-compact low-rise (LR6_Mid410, $H^* = 0.06$ — left column) and high-rise (HR36_Mid410, $H^* = 0.36$ — right column) typologies. Spatial axes refer to the coordinate system defined in Fig. A.2, with the radial axis matching with the $(O, z)$ direction (unfolded view).
<figure><img src="image_32.png" /><figcaption>(c) $H^* = 0.06$, $k = 5$ - $EV_5 = 2.49\%$</figcaption></figure>

<figure><img src="image_33.png" /><figcaption>(e) $H^* = 0.06$, $k = 6$ - $EV_6 = 1.70\%$</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>(b) $H^* = 0.36$, $k = 4$ - $EV_4 = 2.18\%$</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>(d) $H^* = 0.36$, $k = 5$ - $EV_5 = 0.98\%$</figcaption></figure>

<figure><img src="image_36.png" /><figcaption>(f) $H^* = 0.36$, $k = 6$ - $EV_6 = 0.97\%$</figcaption></figure>

<figure><img src="image_37.png" /></figure>

Figure A.11 Spatial modes $v_{4-6}(x)$ and related expansion coefficients $a_{4-6}(t)$ for the semi-compact low-rise (LR6_Mid410, $H^* = 0.06$ — left column) and high-rise (HR36_Mid410, $H^* = 0.36$ — right column) typologies. Spatial axes refer to the coordinate system defined in Fig. A.2, with the radial axis matching with the $(O, z)$ direction (unfolded view).
<figure><img src="image_38.png" /></figure>

<figure><img src="image_39.png" /></figure>

<figure><img src="image_40.png" /></figure>

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /></figure>

<figure><img src="image_43.png" /></figure>

Figure A.12 Daily shortwave downward irradiance (SWD) received by a point facing upwards on the 8th (a, b), 11th (c, d) and 22nd (e, f) July 2023 in Palaiseau, France, as tabulated in the BSRN empirical dataset (blue curve) and estimated with `htrdr-urban` (red curve). Right-hand graphs focus on a specific snippet of their corresponding plots in the left column (grey window). Error bars at each timestep between measured and simulated data are shown in green.