Flexible Treatment of Radiative Transfer in Complex Urban Canopies for Use in Weather and Climate Models

Abstract

We describe a new approach for modelling the interaction of solar and thermal-infrared radiation with complex multi-layer urban canopies. It uses the discrete-ordinate method for describing the behaviour of the radiation field in terms of a set of coupled ordinary differential equations that are solved exactly. The rate at which radiation intercepts building walls and is exchanged laterally between clear-air and vegetated parts of the urban canopy is described statistically. Key features include the ability to represent realistic urban geometry (both horizontal and vertical), atmospheric effects (absorption, emission, and scattering), and spectral coupling to an atmospheric radiation scheme. In the simple case of a single urban layer in a vacuum, the new scheme matches the established matrix-inversion method very closely when eight or more streams are used, but with the four-stream configuration being of adequate accuracy in an operational context. Explicitly representing gaseous absorption and emission in the urban canopy is found to have a significant effect on net fluxes in the thermal infrared. Indeed, we calculate that for the mid-latitude summer standard atmosphere at mean sea level, 37% of thermal-infrared energy is associated with a mean free path of less than 50 m, which is the typical mean line-of-sight distance between walls in an urban area. The interaction of solar radiation with trees has been validated by comparison to Monte Carlo benchmark calculations for an open forest canopy over both bare soil and snow.

Appendices

Appendix 1: List of Symbols

The following list includes symbols used in more than one equation in Sect. 2.

${\mathbf {A}}_{\mathrm {above\,}j-1/2}$ :

Albedo to diffuse downwelling radiation of an entire scene below interface $j-1/2$, with matrix elements configured for regions in the layer above the interface (layer $j-1$)

$c^i_j$ :

Fraction of layer j occupied by region i, which may be clear-air (a), vegetation (v) or building (b)

${\mathbf {D}}_{\mathrm {above}\,j-1/2}$ :

As ${\mathbf {A}}_{\mathrm {above\,}j-1/2}$ but for direct radiation

$e^i_{k}$ :

Rate at which radiation in region i and stream k is extinguished by scattering or absorption, per unit vertical distance travelled (m$^{-1}$)

$e^i_{kl}$ :

Rate at which radiation in region i and stream k is scattered into stream l of the same or the opposite hemisphere ($\hbox {m}^{-1}$)

${\mathbf {E}}_j$ :

Transmission matrix for direct radiation in layer j

$f^{ij}_k$ :

Rate at which radiation in the angle indexed k passes from region i to j per unit vertical distance travelled (m$^{-1}$); if j is ‘w’ then interception by the wall is indicated

$h_k$ :

Weighting of stream k as the contribution to an irradiance into a horizontal surface

$L^{ij}$ :

Length of interface between regions i and j normalized by the area of the domain (m$^{-1}$); if j is ‘w’ then the normalized length of the building walls is indicated

$p^w$ :

Fraction of the reflection from the walls that is specular

${\mathbf {R}}_j$ :

Diffuse reflectance matrix of layer j

$s^i$ :

Element of vector ${\mathbf {s}}$: the direct irradiance in region i ($\hbox {W m}^{-2}$)

${\mathbf {s}}$ :

Vector of downwelling direct irradiances in each region at a particular height, where subscript $j-1/2$ indicates irradiances at interface $j-1/2$, and subscripts ‘above’ or ‘below’ indicate irradiances in the regions just above or below an interface ($\hbox {W m}^{-2}$)

${\mathbf {S}}^+_j$ :

Matrix describing the fraction of direct solar radiation entering each region at the top of layer j that is scattered back up out of each region

${\mathbf {S}}^-_j$ :

Matrix describing the fraction of direct solar radiation entering each region at the top of layer j that is scattered out of each region at the base of that layer

${\mathbf {T}}_j$ :

Diffuse transmittance matrix of layer j

$u^i_k$ :

Element of vector ${\mathbf {u}}$: the irradiance in region i and stream k ($\hbox {W m}^{-2}$)

${\mathbf {u}}$ :

Vector of upwelling irradiances in each region and stream at a particular height ($\hbox {W m}^{-2}$; subscripts as for ${\mathbf {s}}$)

${\mathbf {U}}_{j-1/2}$ :

Upward overlap matrix expressing how upwelling irradiances in regions just below interface $j-1/2$ are transported into regions just above

$v_k$ :

Weighting of stream k as the contribution to an irradiance into a vertical surface

$v^i_k$ :

Element of vector ${\mathbf {v}}$: the irradiance in region i and stream k ($\hbox {W m}^{-2}$)

${\mathbf {v}}$ :

Vector of downwelling diffuse irradiances in each region and stream at a particular height ($\hbox {W m}^{-2}$; subscripts as for ${\mathbf {s}}$)

${\mathbf {V}}_{i-1/2}$ :

Downward overlap matrix expressing how downwelling diffuse irradiances in regions just above interface $j-1/2$ are transported into regions just below

$w_k$ :

Weighting of stream k according to Gaussian quadrature

${\mathbf {W}}_{j-1/2}$ :

As ${\mathbf {V}}_{j-1/2}$ but for downwelling direct irradiances

X :

e-Folding separation distance in an exponential fit to the distribution of wall-to-wall separation distances (m); see Hogan (2019)

z :

Depth into the canopy measured from the top of the tallest building (m)

$\alpha ^i$ :

Albedo of facet i, which may be wall (w), roof (b), ground beneath clear-air (a) or ground beneath vegetation (v)

$\varvec{\Gamma }$ :

Matrix expressing the rates of radiation exchange between irradiance components in each stream and each region (m$^{-1}$)

$\varvec{\Gamma }_0\cdots \varvec{\Gamma }_4$ :

Sub-matrices of $\varvec{\Gamma }$ representing specific interactions (m$^{-1}$)

$\theta _k$ :

Zenith angle of stream k, where $k=0$ indicates the solar zenith angle

$\mu _k$ :

Cosine of $\theta _k$

$\sigma ^i$ :

Extinction coefficient of region i (m$^{-1}$)

$\omega ^i$ :

Single scattering albedo of region i

Appendix 2: The Eigendecomposition Method in the Shortwave

This appendix describes how the matrices listed in Sect. 2.3 are derived from the $m\times m$ matrix $\varvec{\Gamma }$ in (3) for a layer of thickness $\Delta z$. The first step is to decompose $\varvec{\Gamma }$ into eigenvalues $\lambda _k$ and corresponding eigenvectors ${\mathbf {g}}_k$ (for k from 1 to m), such that solutions to (2) have the form

$$\begin{aligned} \left( \begin{array}{c} {\mathbf {u}}\\ {\mathbf {v}}\\ {\mathbf {s}} \end{array}\right) _{\!\!z} =\sum _{k=1}^m c_k{\mathbf {g}}_k\exp \left[ \lambda _k(z-z_{j-1/2})\right] , \end{aligned}$$

(44)

where the $c_j$ coefficients are determined by the boundary conditions. The nature of the matrices in radiative transfer problems is such that the eigenvalues and eigenvectors are always real, making this decomposition more efficient (Stamnes et al. 1988).

Due to the zero elements and block structure of $\varvec{\Gamma }$, the eigenvalues and eigenvectors can be computed efficiently by building them up from eigendecompositions of the smaller sub-matrices. If matrix ${\mathbf {G}}$ is defined such that its kth column contains eigenvector ${\mathbf {g}}_k$ then it has the following form

$$\begin{aligned} {\mathbf {G}}=\left( \begin{array}{ccl} {\mathbf {G}}_1 &{} {\mathbf {G}}_2 &{} {\mathbf {G}}_{3} \\ {\mathbf {G}}_2 &{} {\mathbf {G}}_1 &{} {\mathbf {G}}_{4} \\ &{} &{} {\mathbf {G}}_0 \end{array}\right) . \end{aligned}$$

(45)

The sub-matrix ${\mathbf {G}}_0$, and its corresponding eigenvalues, are computed by performing an eigendecomposition of just the $\varvec{\Gamma }_0$ sub-matrix of (3). The direct transmission matrix ${\mathbf {E}}$ is simply the matrix exponential of $\varvec{\Gamma }_0$ (Hogan et al. 2016), which can be computed directly from the eigendecomposition.

Stamnes et al. (1988) showed that ${\mathbf {G}}_1$ and ${\mathbf {G}}_2$ could be computed by manipulating the result of an eigendecomposition of $(\varvec{\Gamma }_1-\varvec{\Gamma }_2)(\varvec{\Gamma }_1+\varvec{\Gamma }_2)$. If $\varvec{\Gamma }_1$ and ${\mathbf {G}}_1$ are of size $n\times n$, then the first n eigenvalues of ${\mathbf {G}}$ are positive, and the second n are negative with $\lambda _{k+n}=-\lambda _k$. This latter property is exploited in the computation of the diffuse reflectance and transmittance matrices, ${\mathbf {R}}$ and ${\mathbf {T}}$. These can be considered to be the irradiances exiting each side of the layer in response to each element of the downwelling irradiance at the top, ${\mathbf {v}}_{j-1/2}$, being set to unity in turn, while all elements of the upwelling irradiance at the base, ${\mathbf {u}}_{j+1/2}$, are set to zero. The direct irradiance is also zero, so we may simplify the problem by excluding the eigenvectors corresponding to direct radiation held in the right column of sub-matrices in (45). Thus we seek n sets of $c_k$ coefficients from (44), one set for each element of ${\mathbf {v}}_{j-1/2}$. Packing these coefficients into a $2n\times n$ matrix ${\mathbf {C}}$ leads to the following

$$\begin{aligned} \left( \begin{array}{ll} {\mathbf {G}}_1{\mathbf {D}}^{-1} &{} {\mathbf {G}}_2{\mathbf {D}}\\ {\mathbf {G}}_2 &{} {\mathbf {G}}_1 \end{array}\right) {\mathbf {C}}=\left( \begin{array}{cc} {\mathbf {0}}\\ {\mathbf {I}} \end{array}\right) , \end{aligned}$$

(46)

where ${\mathbf {D}}$ is a diagonal matrix with $\exp (-\lambda _k\Delta z)$ on the kth diagonal, and hence ${\mathbf {D}}^{-1}$ is likewise but with $\exp (+\lambda _k\Delta z)$ on the kth diagonal. Each row of (46) expresses (44) for one of the boundary conditions. The top half (i.e. the top n rows) expresses the condition that the upwelling irradiances at the base of the layer are all zero, while the bottom half expresses that the downwelling irradiances at the top are set to one in turn.

The problem with solving (46) computationally is that for very optically thick layers, $\exp (+\lambda _j\Delta z)$ can overflow, even in double precision. Therefore, we follow the stabilization procedure of Stamnes et al. (1988) and solve instead for a scaled set of coefficients ${\mathbf {C}}'$ defined as

$$\begin{aligned} {\mathbf {C}}'={\mathbf {C}}\left( \begin{array}{cc} {\mathbf {D}}^{-1}\\ &{} {\mathbf {I}}\end{array}\right) . \end{aligned}$$

(47)

Thus (46) becomes

$$\begin{aligned} \left( \begin{array}{ll} {\mathbf {G}}_1 &{} {\mathbf {G}}_2{\mathbf {D}}\\ {\mathbf {G}}_2{\mathbf {D}} &{} {\mathbf {G}}_1 \end{array}\right) {\mathbf {C}}'=\left( \begin{array}{cc} {\mathbf {0}}\\ {\mathbf {I}} \end{array}\right) . \end{aligned}$$

(48)

This can be solved efficiently by exploiting the block-symmetric structure of the matrix on the left-hand side, which enables its inverse to be written in terms of the Schur complement. The presence of zeros on the right-hand side then means that not all parts of the inverted matrix need to be computed.

Once we have ${\mathbf {C}}'$, we evaluate the upwelling part of (44) at the top of the layer and the downwelling part at the base of the layer, which are equal to the diffuse reflectance and transmittance matrices,

$$\begin{aligned} \left( \begin{array}{ll} {\mathbf {R}}\\ {\mathbf {T}} \end{array}\right) =\left( \begin{array}{ll} {\mathbf {G}}_1 &{} {\mathbf {G}}_2\\ {\mathbf {G}}_2{\mathbf {D}}^{-1} &{} {\mathbf {G}}_1{\mathbf {D}} \end{array}\right) {\mathbf {C}} =\left( \begin{array}{ll} {\mathbf {G}}_1{\mathbf {D}} &{} {\mathbf {G}}_2\\ {\mathbf {G}}_2 &{} {\mathbf {G}}_1{\mathbf {D}} \end{array}\right) {\mathbf {C}}'. \end{aligned}$$

(49)

The absence of ${\mathbf {D}}^{-1}$ in (48) and (49) shows that, via the use of the scaled set of coefficients ${\mathbf {C}}'$, we can compute ${\mathbf {R}}$ and ${\mathbf {T}}$ without computing any positive exponentials. The matrices ${\mathbf {S}}^+$ and ${\mathbf {S}}^-$, which describe the fraction of incoming direct radiation scattered into the upwelling and downwelling diffuse streams, may be computed using a similar procedure to ${\mathbf {R}}$ and ${\mathbf {T}}$ but instead deriving a set of coefficients consistent with the each element of the direct irradiance at the top of the layer being set to one in turn.

Section 2.5 computes the net radiation absorbed at each facet in layer j from the vertically-integrated irradiances across the layer. Here we describe how to compute the vertically-integrated shortwave irradiances, $\hat{{\mathbf {f}}}_j$, in terms of the irradiances at a given height, ${\mathbf {f}}(z)$. These vectors are simply the concatenation of the individual irradiance vectors,

$$\begin{aligned} \hat{{\mathbf {f}}}_j=\left[ \begin{array}{c} \hat{{\mathbf {u}}}_j\\ \hat{{\mathbf {v}}}_j\\ \hat{{\mathbf {s}}}_j\end{array}\right] \quad \mathrm {and}\quad {\mathbf {f}}(z)= \left[ \begin{array}{c} {{\mathbf {u}}(z)}\\ {{\mathbf {v}}(z)}\\ {{\mathbf {s}}(z)} \end{array}\right] . \end{aligned}$$

(50)

The vertical integral of ${\mathbf {f}}(z)$ is

$$\begin{aligned} \hat{{\mathbf {f}}}_j = \int _{z_{j-1/2}}^{z_{j+1/2}} {\mathbf {f}}(z) \mathrm {d}z, \end{aligned}$$

(51)

which may be evaluated by writing the solution to (2) in terms of a matrix exponential

$$\begin{aligned} {\mathbf {f}}(z) = \exp \left[ \varvec{\Gamma }\times (z-z_{j-1/2})\right] {\mathbf {f}}_{j-1/2}, \end{aligned}$$

(52)

where ${\mathbf {f}}_{j-1/2}$ is the irradiance vector at the top of the layer, which has already been computed. Substituting into (51) and integrating yields

$$\begin{aligned} \hat{{\mathbf {f}}}_j = \varvec{\Gamma }^{-1}\left[ \exp (\varvec{\Gamma }\Delta z_j)-{\mathbf {I}}\right] {\mathbf {f}}_{j-1/2} \end{aligned}$$

(53)

where $\Delta z_j=z_{j+1/2}-z_{j-1/2}$ is the thickness of the layer. Substituting in (52) at $z=z_{j+1/2}$ yields

$$\begin{aligned} \hat{{\mathbf {f}}}_j = \varvec{\Gamma }^{-1}\left( {\mathbf {f}}_{j+1/2} - {\mathbf {f}}_{j-1/2} \right) . \end{aligned}$$

(54)

Thus we may compute the vertically-integrated irradiances across a layer from $\varvec{\Gamma }$ and the known irradiances at the top and base of the layer.

Appendix 3: The Eigendecomposition Method in the Longwave

In the longwave, solutions to (32) have the form

$$\begin{aligned} \left( \begin{array}{c} {\mathbf {u}}\\ {\mathbf {v}} \end{array}\right) _{\!\!z} = \sum _{k=1}^m c_k{\mathbf {g}}_k\exp \left[ \lambda _k(z-z_{j-1/2})\right] - \varvec{\Gamma }^{-1}\left( \begin{array}{r} \mathbf {-b}\\ {\mathbf {b}} \end{array}\right) , \end{aligned}$$

(55)

where the first term on the right-hand side is the homogeneous part of the solution and is expressed in terms of eigenvalues and eigenvectors just as in the shortwave solution (Eq. 44). The reflectance and transmittance matrices are computed exactly as in the shortwave case described in Appendix 2. We also require ${\mathbf {p}}$, the irradiance upwelling from the top or downwelling from the base of the layer due only to emission within the layer, which may be found by setting boundary conditions that the downwelling radiation at the top and the upwelling radiation at the base of the layer are zero. As in Appendix 2, we need to solve a system of equations to obtain the corresponding scaled set of coefficients,

$$\begin{aligned} \left( \begin{array}{ll} {\mathbf {G}}_1 &{} {\mathbf {G}}_2{\mathbf {D}}\\ {\mathbf {G}}_2{\mathbf {D}} &{} {\mathbf {G}}_1 \end{array}\right) {\mathbf {c}}_b'=\varvec{\Gamma }^{-1}\left( \begin{array}{r} -{\mathbf {b}}\\ {\mathbf {b}} \end{array}\right) , \end{aligned}$$

(56)

where now we only need one set of coefficients contained in vector ${\mathbf {c}}_b'$, and the inhomogeneous term from (55) now appears on the right-hand side. Once the coefficients have been computed, the upwelling irradiance at the top of the layer is equal to ${\mathbf {p}}$, given by the top half of (55) in matrix form

$$\begin{aligned} {\mathbf {p}}=\left( \begin{array}{cc} {\mathbf {G}}_1{\mathbf {D}}&~~{\mathbf {G}}_2 \end{array}\right) {\mathbf {c}}_b' +\varvec{\Gamma }^{-1}{\mathbf {b}}, \end{aligned}$$

(57)

where as in Appendix 2 we account for the fact that the coefficients in ${\mathbf {c}}_b'$ are scaled.

Finally we compute the layer-integrated longwave irradiances needed in (37). We integrate (55) with height across the layer of thickness $\Delta z$ to obtain

$$\begin{aligned} \left( \begin{array}{c} \hat{{\mathbf {u}}}\\ \hat{{\mathbf {v}}} \end{array}\right) _{\!\!j}&= \sum _{k=1}^m c_k{\mathbf {g}}_k\frac{\exp \left( \lambda _k\Delta z\right) -1}{\lambda _k} -\mathbf {\varvec{\Gamma }}^{-1} \left( \begin{array}{rr}-{\mathbf {b}}\\ {\mathbf {b}}\end{array}\right) \Delta z\nonumber \\&= {\mathbf {G}}\left( \begin{array}{cc}{\mathbf {Z}}\\ &{}{\mathbf {Z}}\end{array}\right) {\mathbf {c}}' -\mathbf {\varvec{\Gamma }}^{-1} \left( \begin{array}{rr}-{\mathbf {b}}\\ {\mathbf {b}}\end{array}\right) \Delta z, \end{aligned}$$

(58)

where the first term on the second line has been written in terms of a vector of scaled coefficients ${\mathbf {c}}'$, and ${\mathbf {Z}}$ is a diagonal matrix whose kth diagonal element is $[1-\exp (-\lambda _k\Delta z)]/\lambda _k$. The coefficients ${\mathbf {c}}'$ are the sum of the contribution from radiation emitted within the layer, ${\mathbf {c}}_b'$, radiation entering from above, ${\mathbf {C}}'{\mathbf {v}}_{j-1/2}$, and radiation entering from below, ${\mathbf {C}}'{\mathbf {u}}_{j+1/2}$ (the latter being prefixed by a term to swap the elements of ${\mathbf {C}}'$ since the coefficients in this matrix were derived for radiation entering from above),

$$\begin{aligned} \left( \begin{array}{c} \hat{{\mathbf {u}}}\\ \hat{{\mathbf {v}}} \end{array}\right) _{\!\!j} = {\mathbf {G}}\left( \begin{array}{cc}{\mathbf {Z}}\\ &{}{\mathbf {Z}}\end{array}\right) \left[ {\mathbf {C}}'{\mathbf {v}}_{j-1/2} + \left( \begin{array}{cc}&{}{\mathbf {I}}\\ {\mathbf {I}}\end{array}\right) {\mathbf {C}}'{\mathbf {u}}_{j+1/2} + {\mathbf {c}}'_b\right] -\mathbf {\varvec{\Gamma }}^{-1} \left( \begin{array}{rr}-{\mathbf {b}}\\ {\mathbf {b}}\end{array}\right) \Delta z. \end{aligned}$$

(59)