# Higher-order decomposition of the urban solar resource variability in space and time

Guillaume Le Gall, Martin Thebault, Julien Ramousse

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Guillaume Le Gall, Martin Thebault, Julien Ramousse. Higher-order decomposition of the urban solar resource variability in space and time. Journal of Physics: Conference Series, 2025, 3140, pp.032004. 10.1088/1742-6596/3140/3/032004. [https://hal.science/hal-05386114v1](https://hal.science/hal-05386114v1)

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Submitted on 27 Nov 2025

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# Higher-order decomposition of the urban solar resource variability in space and time

G F J Le Gall, M Thebault and J Ramousse

LOCIE UMR 5271, Université Savoie Mont Blanc, CNRS, 60 avenue du Lac Léman, Le Bourget-du-Lac, 73370, France

E-mail: guillaume.le-gall@univ-smb.fr

**Abstract.** The three-dimensional urban geometry introduces complex variations in incident solar radiation across multiple spatiotemporal scales. This study relies on multilinear modal decomposition for characterising the simultaneous variability of the urban solar resource in space and time. The annual irradiance on building envelopes in a theoretical district granted structural and material heterogeneity is estimated using a backward Monte Carlo method. The simulated field is arranged as a three-way tensor from its known temporal characteristics and decomposed using Higher-Order Singular Value Decomposition (HOSVD). Dominant spatial and temporal (daily and seasonal) modes of variation are extracted and ranked according to their influence, enabling the identification of main variability-prone intra-urban regions. The approach provides insights into specific periods of heightened irradiance variations throughout the year and day, while revealing corresponding disruptions to its spatial distribution. The impact of prominent features on the direct sunlight obstruction is especially portrayed.

## 1. Introduction

As urbanisation accelerates and energy demand shifts toward cities [1], planning policies assume more sustainable approaches to reduce carbon footprints, and transition to more comfortable and energy-efficient built areas [2]. Given its renewable and abundant nature, solar energy production on the urban scale plays a key role in optimising resource management and operational efficiency.

This is yet challenged by the inherent intricacy the urban fabric is granted [3]. The morphological heterogeneity of three-dimensional structures (e.g. shape, size, orientation) together with the optical and radiative diversity of encountered materials (e.g. spectral and diffuse reflectivity, absorptivity) largely disrupt the direct and diffuse components of incident radiation [1, 4]. This especially induces dynamic overshadowing effects and multiple interreflections on urban surfaces, in turn causing important spatiotemporal variations of the solar radiative flux density (i.e. irradiance) received by any intra-urban region. These further add to stochastic fluctuations from radiative events occurring within the atmospheric layer, and deterministic cyclic variations due to the apparent movement of the sun.

Urban irradiance eventually varies across a wide range of spatial (sub-metres to multiple kilometres) and temporal (seconds to many days and months) scales [5]. This rapidly prevents its fine characterisation in both space and time using simple models or direct observation [6]. It is yet needed at many levels, e.g. definition of robust climate forecasting models [7], reliable management of energy networks [1] from an optimal positioning of solar collectors. Until recent works by the authors [8, 9], no attempt had been done to investigate the variation of irradiance

<figure><img src="image_2.png" /></figure>
in an urban context with highly-resolved space and time resolutions. While some research has adopted high temporal sampling rates [10, 11], no simultaneous attention has been paid to sub-kilometre changes. Three-dimensional urban features, omitted in such approaches, yet significantly impact the local distribution of irradiance on the metropolitan scale.

This work uses multilinear low-rank approximation to uncover dominant urban features on the variability of irradiance at an obstacle-resolving degree, with a sub-hour resolution. The study is conducted on a mid-high latitude theoretical aggregate of cuboid buildings, granted morphological and surface material heterogeneity (Section 2.1). Independent backwards Monte Carlo simulations are performed every 15 minutes of a typical year under clear sky conditions to estimate the shortwave radiative flux density on all building faces with a 1 m<fnref n="2" /> spatial sampling (Section 2.2). The irradiance field is then organised as a three-way array from its *a priori* known cyclic periodicity and expanded in a lower-dimensional space minimising its least-squares reconstruction error by means of Higher-Order Singular Value Decomposition (HOSVD) [12] (Section 2.3). Extracted modes of variation in space and time (daily and seasonal) are presented in Section 3, and their underlying physics is analysed as regards the urban model properties. Section 4 finally discusses the findings and concludes about implications of the approach.

## 2. Methodology

### 2.1. *Physical urban model and atmospheric conditions*

The district geometry defines an arrangement of unglazed parallelepiped structures populating an $L_D \times L_D$ plane ($L_D$ = 100 m), surrounded by a buffer band of width $l_D$ = 0.1$L_D$ (Fig. 1). Buildings location, footprint, height and orientation are randomly assigned for the urban block to display morphological heterogeneity with a global aspect ratio<fnref n="1" /> $H^* = 0.2$ and a total site coverage<fnref n="2" /> $\kappa$ = 0.5 (compact mid-rise typology [13]).

<figure><img src="image_3.png" /><figcaption>**Figure 1.** 3-D (a) and top (b) views of the evaluated district geometry. Building azimuths are measured counter-clockwise from east.</figcaption></figure>

All surfaces are granted grey opaque Lambertian materials, described by their diffuse reflectances $\rho$ (no directional and spectral dependency) whose values have been set accordingly to common actual urban materials [14]. Façades of height $H \le 15$ m are assigned a brick material ($\rho_{20}$ = 0.2), and concrete otherwise ($\rho_{30}$ = 0.3). An exception is made for the walls of two adjacent buildings in the south-west of the district where a high-albedo whitewash paint ($\rho_{60}$ = 0.6) is specifically selected (B1 and B6 in Fig. 1.b). Tiled roofs (reflectance $\rho_{20}$) are chosen. Patterns of same reflectance have been randomly distributed across the district to mimic grass (green patches), so that the pervious surface fraction<fnref n="3" /> equals $\kappa_p$ = 0.15. An asphalt pavement ($\rho_{10}$ = 0.1) is specified for the remaining ground (including the buffer band). By default, the average ground-level temperature for the chosen location (see Section 2.2) is set for all surfaces.

<footnote n="1">¹ Ratio of average building height $\bar{H}$ to the width $L_{100}$ of a reference 100 × 100 m² plane (valid for $H \le 100$ m).</footnote>

<footnote n="2">² Ratio of the cumulative building footprints to the effective site area (excluding the buffer band overtaking).</footnote>

<footnote n="3">³ Ratio of the pervious ground cover to the total ground surface area (excluding the buffer band overtaking) [3].</footnote>
The geometry and surface diffuse reflectances eventually set boundary conditions with the atmosphere. Clouds are omitted from the calculation, as the investigation focuses on the specific variability induced by urban elements. The ecRad radiation scheme for the European Centre for Medium-Range Weather Forecasts (ECMWF) [15] is used to describe the volumetric atmosphere in the absence of suspended water droplets, from vertical profiles of its physical and radiative properties for the selected location of Paris, France (mid-high latitude temperate climate).

## 2.2. Annual irradiance simulation

Empirical measurements of urban irradiance on an obstacle-resolving scale over an entire year is particularly challenging, and justifies the current lack of such public datasets. A numerical approach is adopted here to estimate the distribution of solar radiative flux in the physical model on fine spatial and temporal scales. The reference model of urban radiative transfer htrdr-urban is employed. The code has undergone rigorous validation against a radiosity-based benchmark solution [4], and has recently been applied in research involving urban geometries [8, 16]. It leverages Monte Carlo methods (MCM) to solve the integral form of the radiative transfer equation within a backwards ray tracing framework from tabulated solar spectral irradiance data [17]. By integrating null-collision algorithms, the model remains quasi-insensitive to the description of the three-dimensional scene [4] while accurately capturing key phenomena (e.g. multi-reflections, scattering), thus mitigating challenges posed by limited empirical data.

The annual shortwave radiative flux density in the range [0.3 µm; 3.0 µm] on the envelope of all ten buildings in Fig. 1 is evaluated. Each surface is sampled in a two-dimensional regular grid of resolution $\Delta x = 1$ m ($n_x = 28,135$ evaluation points). Independent MCM calculations are then performed for an infinite repetition of the district geometry along both (O, x) and (O, y) axes every $\Delta t_y = 4$ days of a typical non-leap year ($n_{t_y} = 92$ days) with a $\Delta t_d = 15$ min daily timestep ($n_{t_d} = 96$ quarter-hour intervals) from January 1st, 00:00, so that $n_t = n_{t_y} \times n_{t_d} = 8,832$ is the total number of point-in-time estimations. A ray tracing resolution of 10,000 samples per pixel is set. Solar azimuth and elevation angles are obtained from the NREL solar position algorithm [18] for the site geographic coordinates (48°51′0″N; 2°20′24″E; 49m alt.) and average weather conditions at ground level ($P = 1013$ Pa; $T = 294.2$ K), with an uncertainty of $\pm 0.0003°$. Note that values of P and T remain constant across successive MCM estimations.

## 2.3. Higher-Order Singular Value Decomposition

Let $e(\mathbf{x}, t)$ be the surface field of irradiance received by the ensemble of building facets $\Omega \subset \mathbb{R}^3$ over the annual period $T \subset \mathbb{R}$. To isolate variations exclusively driven by urban features, the deterministic contribution of the natural sun path is subtracted, leading to:

$$e(\mathbf{x}, t) = \tilde{e}(\mathbf{x}, t) - e_0(\mathbf{x}, t), \quad \mathbf{x} \in \mathbb{R}^3, t \in T, \qquad (1)$$

with $\tilde{e}(\mathbf{x}, t)$ the radiative flux density observed on $T \otimes \Omega$ in the presence of obstacles and $e_0(\mathbf{x}, t)$ the amount received on the same grid in an equivalent obstruction-free context. Subspaces $T = \{t_{k_{12}}\}_{k_{12}=1}^{n_t}$ and $\Omega = \{\mathbf{x}_{k_3}\}_{k_3=1}^{n_x}$ are sampled accordingly to the spatiotemporal definition of independent MCM simulations (Section 2.2). A priori known cyclic modulations of irradiance are further considered from a distinction of annual and daily time coordinates in two separate vector spaces $T_y = \{t_{k_1}\}_{k_1=1}^{n_{t_y}}$ and $T_d = \{t_{k_2}\}_{k_2=1}^{n_{t_d}}$, respectively, so that $T = T_y \otimes T_d$.

The process $e(\mathbf{x}, t)$ is thus expressed as a three-way array $\mathcal{E} \in \mathbb{R}^{n_{t_y} \times n_{t_d} \times n_x}$ of multilinear rank ($R_1, R_2, R_3$) and expanded in an orthonormal basis of vector outer products $(\mathbf{v}_{r_1}^{(1)} \otimes \mathbf{v}_{r_2}^{(2)} \otimes \mathbf{v}_{r_3}^{(3)})$ ($r_i \in \{1, 2, \dots, R_i\}, i = 1, 2, 3$) by Higher-Order Singular Value Decomposition (HOSVD) [12]:

$$\hat{\mathcal{E}} = \sum_{r_1=1}^{R_1} \sum_{r_2=1}^{R_2} \sum_{r_3=1}^{R_3} \sigma_{r_1 r_2 r_3} (\mathbf{v}_{r_1}^{(1)} \otimes \mathbf{v}_{r_2}^{(2)} \otimes \mathbf{v}_{r_3}^{(3)}). \qquad (2)$$
Eigenbases {$\mathbf{v}_{r_1}^{(1)}$}$_{r_1=1}^{R_1}$, {$\mathbf{v}_{r_2}^{(2)}$}$_{r_2=1}^{R_2}$ and {$\mathbf{v}_{r_3}^{(3)}$}$_{r_3=1}^{R_3}$, respectively spanning $T_y$, $T_d$ and $\Omega$, are orthonormal and sought to solve the least-squares approximation problem:

$$ \min_{\hat{\mathcal{E}}} \|\mathcal{E} - \hat{\mathcal{E}}\|_F, \qquad (3) $$

for any rank-$(r_1, r_2, r_3)$ truncation of $\hat{\mathcal{E}}$, where the Frobenius norm $\|\cdot\|_F$ is reasonably extended to multilinear algebra. HOSVD eventually partitions the multivariate variability in $\mathcal{E}$ from its exact factorisation into univariate modes of variation $\mathbf{v}_{r_i}^{(i)}$ along all three dimensions $i = 1, 2$ and $3$, i.e. time (daily and quarter-hourly) and space, respectively. Core entries $\sigma_{r_1r_2r_3}$ quantify intermodal interactions, capturing the proportion of the global dynamics accounted for by individual triplets ($\mathbf{v}_{r_1}^{(1)}, \mathbf{v}_{r_2}^{(2)}, \mathbf{v}_{r_3}^{(3)}$). The cumulative explained variability for first $r_i$ modes along dimension $i$ is therefore given by:

$$ EV_{r_i}^{(i)} = \frac{\sum_{k=1}^{r_i} \sigma_k^{(i)2}}{\sum_{r_1=1}^{R_1} \sum_{r_2=1}^{R_2} \sum_{r_3=1}^{R_3} \sigma_{r_1 r_2 r_3}^2}. \qquad (4) $$

Here, $\sigma_k^{(i)}$ represents the singular value associated with the mode-$k$ component in the way-$i$ unfolding of $\mathcal{E}$, and is conventionally used to order extracted modes by their respective contribution to the dimension-wise variability of the signal.

## 3. Results

The HOSVD decomposition of the space-time irradiance received by building surfaces in Fig. 1 is presented here. The analysis restricts to first three modes $\mathbf{v}_{1-3}^{(i)}$, $i = 1, 2$ and $3$ (Fig. 2). Temporal modes ($i = 1, 2$) obtained for a $1 \text{ m}^2$ horizontal patch in an unobstructed environment (NoObs) are plotted for comparison (dotted lines in Fig. 2). All values represent absolute variations.

### 3.1. Separate description of principal irradiance variations in space and time

Plots of average and cumulative annual irradiances in Fig. 2.a immediately stress the difficulty of simultaneously analysing its spatial and temporal variability. HOSVD provides a separation of space and time variables, and further distinguishes between seasonal and daily variations, as respectively described by modes $\mathbf{v}_{r_3}^{(3)}$, $\mathbf{v}_{r_1}^{(1)}$ and $\mathbf{v}_{r_2}^{(2)}$ (Fig. 2.b-d). The latter successively picture principal changes in the global irradiance along their respective dimension $i = 1, 2$ and $3$ (in the narrow sense of the Frobenius norm $\|\cdot\|_F$). Slow variations occurring on large periods prevail here. Expected average annual and daily irradiance evolutions, differentiating peak exposure periods (e.g. equinoxes, summer solstice, midday) from lower-exposure times (e.g. winter solstice, early morning, late afternoon), are respectively depicted by $\mathbf{v}_1^{(1)}$ and $\mathbf{v}_1^{(2)}$ in Fig. 2.b. Similarly, $\mathbf{v}_1^{(3)}$ portrays what can be observed as changes in the annual cumulative irradiance across the district grid (c.f. Fig. 2.a). Higher-frequency refinements emerge in following modes $\mathbf{v}_{2-3}^{(i)}$, which introduce dimension-wise modulations of dominant vectors at levels $r_i = 1$. For instance, the seasonal modulation and morning/afternoon sharing are now clearly described by $\mathbf{v}_3^{(1)}$ (Fig. 2.d) and $\mathbf{v}_2^{(2)}$ (Fig. 2.c), respectively. As regards the spatial distribution of irradiance, finer regional patterns appear from $r_3 = 2$, with local façade variations distinctly captured by $\mathbf{v}_3^{(3)}$ (Fig. 2.d).

It is observed from the distribution of $\sigma_{r_1r_21}$ (Fig. 2.b) that leading modes $\mathbf{v}_1^{(i)}$ along all three ways $i = 1, 2$ and $3$ are strongly linked, and can thus be jointly interpreted. Off-diagonal non-zero elements reveal the additional connections of $\mathbf{v}_1^{(3)}$ with couples ($\mathbf{v}_2^{(1)}, \mathbf{v}_1^{(2)}$) and ($\mathbf{v}_2^{(1)}, \mathbf{v}_3^{(2)}$). Namely, the spatial variability distribution described in Fig. 2.b is still modulated by seasonal and sub-daily variations in Fig. 2.c and d. Similar comments hold for subsequent modes $\mathbf{v}_{2-3}^{(3)}$ as shown by plots of $\sigma_{r_1r_22}$ (Fig. 2.c) and $\sigma_{r_1r_23}$ (Fig. 2.d).
<figure><img src="image_4.png" /><figcaption>**Figure 2.** (a): Average and cumulative annual irradiance on building envelopes in Fig. 1. (b-d): Corresponding HOSVD modes $\mathbf{v}_{1-3}^{(i)}$ ($i = 1, 2, 3$), and first 100 core entries $\sigma_{r_1 r_2 1-3}$.</figcaption></figure>

The cumulative explained variability met by first three modes $\mathbf{v}_{1-3}^{(i)}$ is substantially higher for annual variations ($i = 1, EV_3^{(1)} = 0.91$) than for the two other ways $i = 2$ ($EV_3^{(2)} = 0.72$) and $i = 3$ ($EV_3^{(3)} = 0.70$). Especially, the annual amplitude of irradiance is thus likely to be less impacted by the urban fabric than its daily variations. Overall, leading eigenmodes rapidly explain the dimension-wise dynamics. A 95% description is reached by the first 6/92 (6.5%), 14/96 (14.6%) and 51/28, 135 (0.2%) eigenmodes along ways $i = 1, 2$ and $3$, respectively.

## 3.2. Characterisation of variability-prone urban features

Due to the pre-processing of $e(\mathbf{x}, t)$ in Eq. (1), the magnitude of modes $\mathbf{v}_{r_i}^{(i)}$ is expected to grow proportionally to the disruption of local irradiance by urban features. The urban contribution is confirmed by the extraction of un-noisy temporal coefficients $\mathbf{v}_{r_{1-2}}^{(1-2)}$ and non-null patterns $\mathbf{v}_{r_3}^{(3)}$ in Fig. 2.b-d. Year-long unobstructed zones, e.g. roofs and upper façade regions of high-rise buildings, present no variability (blank patches in $\mathbf{v}_{r_3}^{(3)}$, Fig. 2.b-d). Conversely, surfaces the most impacted by the presence of nearby geometries, e.g. roof of the low-rise cuboid B4 or facing walls of adjacent alike buildings B9 and B10, are immediately highlighted by the dominant mode $\mathbf{v}_1^{(3)}$ (Fig. 2.b). The shading cast on buildings in the north-eastern quarter of the model by surrounding high-rise structures is then clarified by the phase shift in $\mathbf{v}_2^{(3)}$ (Fig. 2.c). This suggests the prevalence of these specific obstructions in the district-wise irradiance variability, which is in turn observed to be particularly important in this region of the model. Note also here the influence of the repeated district geometry on the flux variability for the roof of B4. Facet specific disruptions are further explained by $\mathbf{v}_3^{(3)}$ (Fig. 2.d). Most-disturbing average
shadows cast by the urban geometry on building faces start to be portrayed here, with the projected footprint of nearby obstructions becoming visible. Note the significant shading on B4 mainly induced by the high structure on its south side (B3) being particularly emphasised by $\mathbf{v}_3^{(3)}$. The impact of urban obstacles is also described by temporal modes, from their respective divergences with the unobstructed reference (NoObs). Modulation of the global irradiance amplitude especially occurs here on a daily scale, e.g. see $\mathbf{v}_1^{(2)}$ (Fig. 2.b) and $\mathbf{v}_2^{(2)}$ (Fig. 2.c).

Overall, variations in surface reflectances remain dimly visible in extracted HOSVD modes. As it could have been expected, highly-diffusive buildings B1 and B6 still seem to smooth variability patterns (see $\mathbf{v}_3^{(3)}$ in Fig. 2.d), although no major change is observed here.

**4. Discussion and Conclusions**

In this study, a multilinear low-rank approximation of the shortwave radiative flux density modelled with a backwards Monte Carlo method on building envelopes of a representative district granted a morphological and material heterogeneity has been performed via HOSVD.

Findings demonstrate the benefits of higher-order factorisation like HOSVD in analysing the spatiotemporal characteristics of multidimensional urban irradiance. Prior knowledge of the field specificities is leveraged to systematically decompose the global dynamics into successive modes along expected variations over time (seasonal and daily cycles) and space. This structured approach enables an independent examination of each variable, facilitating the detection of district and façade regions experiencing specific temporal and spatial variations of the solar resource. The subdivision of the temporal dimension enhances the understanding of overall variability in time. Seasonal and daily trends are effectively captured through temporal modes, which successively describe specific frequency ranges, thus pinpointing periods of important irradiance variability on both daily and annual scales. Main related disruptions to the district-wise distribution of solar radiation are uncovered by spatial modes, which provide further insight into the obstruction of direct sunlight and the dynamic shadow patterns created by nearby structures. Especially, most disturbing geometries and intra-urban areas particularly prone to variation of irradiance, such as densely built high-rise zones, are detected. Assuming an appropriate physically-informed arrangement, such approach proves particularly effective for interpreting the variability of complex multidimensional signals evolving across broad spatiotemporal scales like urban irradiance. It becomes possible to assess the individual contribution of each dimension to the global variability while having a precise idea of their mutual connections. However, these interactions between different modes introduce multiple layers of interpretation for each component, which may, in turn, hinder the global analysis.

Driving away from cuboid shapes would lead to more intricate shadowing effects, likely requiring an examination up to a higher level $r_i$. A detailed surface and volume description of material properties, now accounting for their directional and spectral dependencies, is expected to alter the current propagation of radiation within the model and influence the HOSVD analysis. Especially, noise might appear in extracted modes from increased specular reflections between glazed façades and/or volumetric scattering in non-opaque materials. The currently difficult detection of subtle diffuse radiative variations may yet be enhanced. The study being restricted to shortwave radiation, materials temperature has no direct impact on the MCM outcome. While a cloudy atmosphere would introduce additional stochastic fluctuations, these are unwanted here as the focus is on the single impact of urban elements. Eventually, any shift in the site properties (e.g. latitude, ambient temperature) might only lead to trivial changes.

Future research would consider more realistic urban models, especially incorporating specular glazings and pervious vegetation. An extension of the decomposition to $N > 3$ dimensions, introducing a spectral sampling of incident radiation, should also be investigated, with potential applications in Urban Heat Island (UHI) studies.
Acknowledgments

This work was supported by the French National Research Agency under the ‘Invesstissements d’Avenir’ [ANR-18-EURE-0016 Solar Academy].

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