Model of Spectral and Directional Radiative Transfer in Complex Urban Canopies with Participating Atmospheres

Abstract

Thermal heat transfers, including solar and infrared radiation in cities, are key processes for studying urban heat islands, outdoor human thermal comfort, energy consumption, and production. Thus, accurate radiative transfer models are required to compute the solar and infrared fluxes in complex urban geometry accounting for the spectral and directional properties of the atmosphere and city fabric materials. In addition, these reference models may be used to evaluate existing parametrization models of radiative heat transfer and to develop new ones. The present article introduces a new reference model for outdoor radiative exchange based on the backward Monte Carlo method. The integral formulations of the direct and scattered solar, and the terrestrial infrared radiative flux densities are presented. This model can take into account the ground (e.g., roads, grass), different types of buildings and vegetation (e.g., trees consisting of opaque leaves and trunks) with their spectral and directional (Lambertian and specular) reflectivity of materials. Numerical validations of the algorithm are presented against the results of a state-of-the-art model based on the radiosity method for the particular case of an infinitely long street canyon. In addition, the convergence of urban solar radiation budgets is studied for a selection of urban complex geometries including or not the window glazing. Good convergence is found for all geometries, even in the presence of rare events due to specular reflections.

Appendices

Appendix 1: Principles of the Monte Carlo Method to Solve Integrals

The MCM is a stochastic numerical technique to compute integrals based on using sequences of random numbers (Dupree and Fraley 2002; Delatorre et al. 2014; Howell et al. 2020). Let S be the integral of a function f over a multiple dimension domain ${\mathcal {D}}$ (Eq. 23). The steps to compute S via the MCM are as follows:

• to introduce a non-zero PDF p over ${\mathcal {D}}$ (Eq. 23),

• to formulate S as the expectation of W $=$ $\frac{f(X)}{p(X)}$, where X is a random variable distributed according to p,

• to sample numerically a large number N of realizations $x_1$, $x_2$,..., $x_N$ of the random variable X, according to the selected PDF p, and with a random number r generated with a uniform PDF in [0; 1) (Eq. 24),

• to compute the MCM weight $W_k$ $=$ $\frac{f(x_k)}{p(x_k)}$ for each realization $x_k$,

• to retain the mean value for numerical estimation $\tilde{S}$ of S (Eq. 25),

$$\begin{aligned} S= & {} \int _{D} f(x) \;dx = \int _{D} p(x)\, \frac{f(x)}{p(x)}\; dx = E\bigg (\frac{f(X)}{p(X)}\bigg ) = E(W), \end{aligned}$$

(23)

$$\begin{aligned} r= & {} \int _{-\infty }^{x} p(x) \;dx, \end{aligned}$$

(24)

$$\begin{aligned} S \approx \tilde{S}= & {} \frac{1}{N} \sum _{k=1}^{N}W_k(x_k). \end{aligned}$$

(25)

As an example, let’s compute the total emissive power of an opaque surface M and chose $p_\varOmega$ (Eq. 14) as the PDF. In this context of radiative transfer, generic Eq. 23 is replaced by:

$$\begin{aligned} M(\mathbf {y})= & {} \int _{2\pi } \varepsilon '(\mathbf {y},{\omega })\;I_b(\mathbf {y})\; |\mathbf {n}\cdot {\omega }| \;d\varOmega ({\omega }):\\= & {} \int _{2\pi } p_\varOmega ({\omega }) \; \frac{ \varepsilon '(\mathbf {y},{\omega })\;I_b(\mathbf {y})\; |\mathbf {n}\cdot {\omega }|}{p_\varOmega ({\omega }) }\; d\varOmega ({\omega }) ,\\= & {} \int _0^{2\pi } p_\varphi \; d\varphi \int _0^{\frac{\pi }{2}} p_\theta \;d\theta \; \{W\}, \end{aligned}$$

with $\varepsilon '$ the surface directional emissivity and where W is the random variable of the MCM weight computed with the realization $\theta _i$ and $\varphi _i$ of their corresponding random variable following their PDF, $p_\theta$ and $p_\varphi$, respectively. These PDFs and the variable realizations are obtained using a representation of the solid angle with spherical coordinates ($\theta$ is the normal angle and $\varphi$ the azimuth angle):

$$\begin{aligned} p_\varOmega \; d\varOmega= & {} p_\varOmega \; |\sin \theta | \; d\theta \; d\varphi = p_\theta \; d\theta \; p_\varphi \; d\varphi , \end{aligned}$$

and using Eq. 24, a realization of $\theta$ and $\phi$ may be obtained:

$$\begin{aligned} \theta _k= & {} \sin ^{-1}\sqrt{r_k}, \\ \varphi _k= & {} 2\,\pi \,r_k. \end{aligned}$$

The MCM weight saved at each realization has the following expression:

$$\begin{aligned} W_k=\frac{ \varepsilon '(\mathbf {y},{\omega })\;I_b(\mathbf {y})\; |\cos \theta _k \, \sin \theta _k|}{p_\varphi \, p_\theta } = \pi \;\varepsilon '(\mathbf {y},{\omega }(\theta _k,\varphi _k))\;I_b(\mathbf {y}). \end{aligned}$$

The sampling of $\theta _k$ and $\varphi _k$ allows one to compute $W_k$. Evaluating a large number of times (N times) the MCM weight computation and then averaging their values (Eq. 25) will produce a MCM estimate ($\tilde{M}$) of the emissive power.

To evaluate the convergence of a MCM estimate $\tilde{S}$, an associated standard error value $\tilde{\sigma }_{\tilde{S}}$ may be computed with the standard deviation of the MCM weight ($\sigma _W$) based on the statistically independent samples of the N realizations:

$$\begin{aligned} \tilde{\sigma }_{\tilde{S}} = \frac{\sigma _W }{\sqrt{N}}, \end{aligned}$$

where,

$$\begin{aligned} \sigma _W = \sqrt{ \bigg ( \frac{1}{N} \sum _{k=1}^{N} W^2_k\bigg ) - \tilde{S}^{2} }. \end{aligned}$$

As a rule of thumb, a MCM estimate with an associated standard error verifying $\frac{3\,\tilde{\sigma }_{\tilde{S}}}{\tilde{S}}$ $\le$ 0.01 is considered converged for most applications in radiative transfer. However, this rule fails when the standard error shows a low value and the computation is not converged because rare events are not taken into account. This risk may occur if N is too small to sample rare events, whereas they can contribute significantly to the estimate.

Appendix 2: Monte Carlo Estimates, Weights, and Standard Errors

The MCM estimate, weight, and standard error for the direct solar radiative flux density are given by:

$$\begin{aligned} \tilde{\dot{q}}_d({\mathbf {x}}_0)= & {} \frac{1}{N} \sum _{k=1}^{N} W_{\dot{q}_d,k}, \end{aligned}$$

(26)

$$\begin{aligned} W_{\dot{q}_d,k}= & {} {\mathcal {H}}({\mathbf {x}}_{d}\in \partial {\mathcal {D}}_d) \,{\mathcal {H}}(\ell _{e} > \Vert {\mathbf {x}}_0-{\mathbf {x}}_{TOA}\Vert )\,\frac{I_{d,ex} \,|{\mathbf {n}}({\mathbf {x}}_0)\cdot \varvec{\omega }_d|}{p_{\varLambda _m} \; p_{\varLambda } \;p_{\varOmega _d}} ,\nonumber \\ \tilde{\sigma }_{\tilde{\dot{q}}_d}= & {} \frac{1}{\sqrt{N}} \sqrt{\frac{1}{N} \sum _{k=1}^{N}W^2_{\dot{q}_d,k} - \tilde{\dot{q}}_d^2 } . \end{aligned}$$

(27)

The MCM estimate, weight, and standard error for the diffuse solar radiative flux density are given by:

$$\begin{aligned} \tilde{\dot{q}}_{sc}({\mathbf {x}}_0)= & {} \frac{1}{N} \sum _{k=1}^{N} W_{\dot{q}_{sc},k}, \end{aligned}$$

(28)

$$\begin{aligned} W_{\dot{q}_{sc},k}= & {} \frac{|{\mathbf {n}}({\mathbf {x}}_0)\cdot \varvec{\omega }_0| \; I_{d,ex}}{p_{\varLambda _m}\; p_{\varLambda } \;p_{\varOmega }\; p_{\varOmega _d}} \bigg ( \prod _{j=1}^{n_{r}} \bigg [ {\mathcal {H}}({\mathbf {x}}_{j}\in \partial {\mathcal {D}}_{{\mathcal {S}}_L})\,\rho ^{\prime \cap }({\mathbf {x}}_{j},\lambda ) \\& +{\mathcal {H}}({\mathbf {x}}_{j}\in \partial {\mathcal {D}}_{{\mathcal {S}}_F})\,\rho _F({\mathbf {x}}_{j},\lambda ) \bigg ] \bigg ) \bigg ( \prod _{j=1}^{n_{sc}} {\mathcal {H}}(\ell _a>\ell _{j}) \bigg ) \nonumber \\&\bigg (\sum _{j=1}^{n_{sc}} {\mathcal {H}}({\mathbf {x}}_{j}\in {\mathcal {D}}) \, {\mathcal {H}}({\mathbf {x}}_{d}\in \partial {\mathcal {D}}_d) \,{\mathcal {H}}(\ell _{e}> \Vert {\mathbf {x}}_j-{\mathbf {x}}_{TOA}\Vert )\, p_{i} ({\mathbf {x}}_{j},\varvec{\omega }_{d}|\varvec{\omega }_{j-1},\lambda ) \nonumber \\&+\,\sum _{j=1}^{n_{r}} {\mathcal {H}}({\mathbf {x}}_{j}\in \partial {\mathcal {D}}_{{\mathcal {S}}_L}) \, {\mathcal {H}}({\mathbf {x}}_{d}\in \partial {\mathcal {D}}_d) \,{\mathcal {H}}(\ell _{e}> \Vert {\mathbf {x}}_j-{\mathbf {x}}_{TOA}\Vert )\, p_{L} ({\mathbf {x}}_{j},\varvec{\omega }_{d}) \nonumber \\&+\,{\mathcal {H}}({\mathbf {x}}_{n_{r}}\in \partial {\mathcal {D}}_{{\mathcal {S}}_F}) {\mathcal {H}}({\mathbf {x}}_{d}\in \partial {\mathcal {D}}_d) \,{\mathcal {H}}(\ell _{e} > \Vert {\mathbf {x}}_j-{\mathbf {x}}_{TOA}\Vert )\, p_{\varOmega _d} \bigg ),\nonumber \\ \tilde{\sigma }_{\tilde{\dot{q}}_{sc}}= & {} \frac{1}{\sqrt{N}} \sqrt{\frac{1}{N} \sum _{k=1}^{N}W^2_{\dot{q}_{sc},k} - \tilde{\dot{q}}_{sc}^2 } .\nonumber \end{aligned}$$

(29)

In Eq. 29, the total number of events was separated into scattering ($n_{sc}$) and reflection ($n_r$) events.

The MCM estimate, weight and standard error for the total infrared radiative flux density are given by:

$$\begin{aligned} \tilde{\dot{q}}_{ir}({\mathbf {x}}_0)= & {} \frac{1}{N} \sum _{k=1}^{N} W_{\dot{q}_{ir},k} , \end{aligned}$$

(30)

$$\begin{aligned} W_{\dot{q}_{ir},k}= & {} \frac{{\pi } \,I_{b}\big (T({\mathbf {x}}_{j+1}),\lambda \big ) }{p_{\varLambda _m}^{ir}\; p_{\varLambda } } \,{\mathcal {H}}({\mathbf {x}}_{j+1}\in {\mathcal {D}} \cup \partial {\mathcal {D}}_{{\mathcal {S}}_L}\cup \partial {\mathcal {D}}_{{\mathcal {S}}_F}),\\ \tilde{\sigma }_{\tilde{\dot{q}}_{ir}}= & {} \frac{1}{\sqrt{N}}\sqrt{\frac{1}{N}\sum _{k=1}^{N}W^2_{\dot{q}_{ir},k} - \tilde{\dot{q}}^2_{ir} },\nonumber \end{aligned}$$

(31)

where ${\mathbf {x}}_{j+1}$ is the final position of the random radiative path considered as the emission location.