A Scale-Adaptive Approach for Spatially-Varying Urban Morphology Characterization in Boundary Layer Parametrization Using Multi-Resolution Analysis

Abstract

Urban morphology characterization is crucial for the parametrization of boundary-layer development over urban areas. One complexity in such a characterization is the three-dimensional variation of the urban canopies and textures, which are customarily reduced to and represented by one-dimensional varying parametrization such as the aerodynamic roughness length $z_{0}$ and zero-plane displacement $d$. The scope of the paper is to provide novel means for a scale-adaptive spatially-varying parametrization of the boundary layer by addressing this 3-D variation. Specifically, the 3-D variation of urban geometries often poses questions in the multi-scale modelling of air pollution dispersion and other climate or weather-related modelling applications that have not been addressed yet, such as: (a) how we represent urban attributes (parameters) appropriately for the multi-scale nature and multi-resolution basis of weather numerical models, (b) how we quantify the uniqueness of an urban database in the context of modelling urban effects in large-scale weather numerical models, and (c) how we derive the impact and influence of a particular building in pre-specified sub-domain areas of the urban database. We illustrate how multi-resolution analysis (MRA) addresses and answers the afore-mentioned questions by taking as an example the Central Business District of Oklahoma City. The selection of MRA is motivated by its capacity for multi-scale sampling; in the MRA the “urban” signal depicting a city is decomposed into an approximation, a representation at a higher scale, and a detail, the part removed at lower scales to yield the approximation. Different levels of approximations were deduced for the building height $\bar{{H}}$ and planar packing density $\lambda _\mathrm{p}$. A spatially-varying characterization with a scale-adaptive capacity is obtained for the boundary-layer parameters (aerodynamic roughness length $z_{0}$ and zero-plane displacement $d$) using the MRA-deduced results for the building height and the planar packing density with a morphometric model; an attribute that is shown to be of great advantage to multi-scale and multi-resolution numerical weather prediction models.

Appendices

Appendix 1

1.1 The Definition of the Scaling Function in the MRA Framework (Carrier of Approximations)

The axioms of MRA postulate the existence of a sequence of nested subspaces, $\ldots V_j \supset V_{j+1} \ldots j\in Z$, where $Z$ denotes the set of integers, of the Hilbert function space $L^{2}(R)$ of continuous time finite energy signals, $R$ denotes the set of real numbers. In addition the axioms state that the translates, $f(t-2^{j}n)$, by $2^{j}n$ of a signal $f(t) \in V_j$, belong to $V_j$, and its dilation $\frac{1}{\sqrt{2}}f\left( {\frac{t}{2}}\right)$ belongs to $V_{j+1}$. The complete list of MRA axioms that can be found in many textbooks, as for example Mallat (1999), Kaiser (2010) and Daubechies (1992), contains another three, rather technical, axioms. It follows a description of the mathematical properties of the scaling function $\phi (t)$.

Through MRA axioms, it can be shown that a scaling function of support $\delta t$, has a pulse-like shape and can be normalized to unit area,

$$\begin{aligned} \int \limits _{-\infty }^\infty {\phi (t)\hbox {d}t=1}. \end{aligned}$$

(20)

Therefore the dilations $\phi _{j,n}$ of the scaling function are supported on intervals of width $2^{j}\tau$, and by Eq. 20 $\phi$ can be used in place of $P$, which means that the inner products $\left\langle {f,\phi _{j,n}}\right\rangle$ are interpreted as samples at scale $2^{j}\tau$.

Since any function that has a pulse-like shape and satisfies Eq. 20 can be used to approximate the impulse function it can furthermore be used to sample a function at a higher scale, as opposed to the actual values sampled at sharp instants by the impulse function. This means that in the integral Eq. 5 the pulse function $P(t)$ can be substituted by a scaled and translated scaling function, $\phi _{j,n}$, to give

$$\begin{aligned} \frac{1}{2^{j}\tau }\int \limits _{t_n -2^{j}\tau }^{t_n +2^{j}\tau } {f(t)} \phi ^{*}_{j,n} (t)\hbox {d}t=\left\langle {f,\phi _{j,n} } \right\rangle . \end{aligned}$$

(21)

Therefore, according to the discussion in the previous paragraph the inner product Eq. 21 corresponds to samples of the function $f(t)$ at $t_n$ at scale $2^{j}\tau$ or to an approximation of the value of $f(t_n )$ at scale $2^{j}\tau$.

Now $f_j$ is the function reconstructed from samples, $\left\langle {f,\phi _{j,n} } \right\rangle$ of $f$ taken at scale $2^{j}\tau$. The sample $\left\langle {f,\phi _{j,n}}\right\rangle$ approximates the value of f at the time $t_n$. The actual basis functions represent the actual shape that the reconstruction takes in the neighbourhood of $f(t_n)$. This shape is determined by the actual shape of the scaling function as it determines how the neighbourhood of $t_n$ is weighted.

In addition since $V_{-1}\supset V_0$, the following functional equation of the scaling function $\phi$, called the dilation equation, can be deduced,

$$\begin{aligned} \phi (t)=\sqrt{2}\sum _n {h_n } \phi (2t-n), \end{aligned}$$

(22)

where the coefficients $h_n$ are given by $h_n =\left\langle {\phi (t),\phi (2t-n)} \right\rangle$ and they can be thought of as a sequence of low-pass filter coefficients. According to the discussion in the previous paragraph $h_n$ are the samples of the scaling function itself at scale $\frac{1}{2}\tau$ i.e. is a measure of self similarity of $\phi$. These coefficients determine the scaling function and they are very important because, (a) they are used to construct the wavelet function, and (b) their convolution with the analyzing signal yields computational analysis and synthesis procedures.

1.2 The Deduction of the Wavelet Function in the MRA Framework (Carrier of Details)

Wavelets in the MRA framework are created through the orthogonal complements of $V_{j+1},\,W_{j+1}$ in $V_j$, in other words, the subspaces of $L^{2}(R)$ containing signals $f$ of $V_j$, that are orthogonal to signals $g$ of $V_{j+1}$, namely,

$$\begin{aligned} W_{j+1} \equiv \left\{ f \in V_j\hbox { such that }\left\langle {f,g} \right\rangle =0 \forall g \in V_{m+1} \right\} , \end{aligned}$$

(23)

which means that $V_j$ can be expressed as the direct product of $V_{j+1}$ and $W_{j+1}$, written as $V_j =V_{j+1} \oplus W_{j+1}$.

As in the case of the approximation spaces $V_j$, each space $W_j$ has an orthonormal basis constituted by translated and dilated versions, $\psi _{j,n}=\frac{1}{\left( {\sqrt{2}} \right) ^{j}}\psi \left( {\frac{t-2^{j}n}{2^{j}}} \right)$ of a function $\psi \in W_o$, called the mother wavelet. A signal $f\in L^{2}(R)$ can be projected on a function $d_j \in W_j$,

$$\begin{aligned} d_j =\sum _n {\left\langle {f,\psi _{j,n} (t)} \right\rangle \psi _{j,n} (t)}. \end{aligned}$$

(24)

Therefore the detail removed from approximation $f_{j-1}$ to obtain $f_j$ at a higher scale is retained by the inner products $\left\langle {f,\psi _{j,n}}\right\rangle$. Since $\psi (t/2)$ belongs to $W_1$ and $W_1$ is the orthogonal complement to $V_1$ in $V_0$ then $\psi (t)$ can be written as,

$$\begin{aligned} \psi (t)=\sqrt{2}\sum _n {g_n \phi (2t-n)}, \end{aligned}$$

(25)

where $g_n$ are considered as coefficients of a differencing (high-pass) filter. These coefficients can be obtained from the averaging coefficients $h_n$ by,

$$\begin{aligned} g_n =(-1)^{n}h_{1-n}. \end{aligned}$$

(26)

In applications, analysis is performed on a sampled signal $f_\mathrm{s}$ assumed to belong to $V_0$ and hence it is set to $f_0$. Then $f_0$ is decomposed up to a level $k$ according to Eq. 10 and reconstructed back according to

$$\begin{aligned} f_0 =f_k +\sum _{\ell =1}^k {d_\ell }. \end{aligned}$$

(27)

Figure 11 depicts diagrammatically this decomposition and reconstruction; the corresponding levels and subspaces are identified. The maximum value of $k$ is dictated by the number of samples of $f_0$ and the actual wavelet used. The functional Eqs. 22 and 25 are satisfied by many different sets of coefficients $h_n$ and $g_n$, meaning that there is a multitude of different mother wavelets and scaling functions.

Fig. 11

The decomposition and reconstruction of a sampled signal $f_{0}$

Appendix 2: The 2-D Formulation of the MRA

Multi-resolution analysis can be defined on $L^{2}(R^{2})$, the space of 2-D finite energy signals $f(x_1,x_2 )$, by taking the tensor product (Mallat 1989),

$$\begin{aligned} V_j^2 =V_j \otimes V_j, \end{aligned}$$

(28)

of the approximation spaces $V_j$ of the 1-D MRA. As a consequence of the tensor product a scaling function, $\phi (x_1,x_2)\in V_0^2$, associated with the 2-D MRA is obtained by the product of the scaling function $\phi (x)$ with itself,

$$\begin{aligned} \phi ^{2}(x_1,x_2 )=\phi (x_1 )\phi (x_2 ). \end{aligned}$$

(29)

Therefore, MRA for $L^{2}(R^{2})$ can similarly be defined by a sequence of nested subspaces $\ldots V_j^2 \subset V_{j+1}^2 \subset V_{j+2}^2 \ldots$. The basis of each $V_j^2$ is given by the translated and dilated versions $\phi _{j,(n_1,n_2 )}^2 =\phi _{j,n_1 } (x_1 )\phi _{j,n_2 } (x_2 ) \quad j, n_1, n_2 \in \mathrm{Z}$ of the scaling function $\phi ^{2}(x_1,x_2)$. The basis of the orthogonal complements, $W_{j+1}^2$, of $V_{j+1}^2$ in $V_j^2$ constitute the 2-D wavelets.

The basis can be constructed from the 1-D basis of $V_0$ and $W_0$ by noting that,

$$\begin{aligned} V_0^2 =V_0 \otimes V_0 =V_1^2 \oplus W_1^2, \end{aligned}$$

(30)

and then by substituting $V_0 =V_1 \oplus W_1$ and exploiting the distributive property of $\oplus$ over $\otimes$ it can be shown that

$$\begin{aligned} W_1^2 =\left( {V_1 \otimes W_1 } \right) \oplus \left( {W_1 \otimes V_1 } \right) \oplus \left( {W_1 \otimes W_1 } \right) , \end{aligned}$$

(31)

and hence it can be deduced that the basis of $W_j^2$ are the dilated and translated versions of the following functions,

$$\begin{aligned} \psi ^{V_0 \otimes W_0 }&= \phi (x_1 )\psi (x_2 ), \end{aligned}$$

(32)

$$\begin{aligned} \psi ^{W_0 \otimes V_0 }&= \psi (x_1 )\phi (x_2 ),\end{aligned}$$

(33)

$$\begin{aligned} \psi ^{W_0 \otimes W_0 }&= \psi (x_1 )\psi (x_2 ), \end{aligned}$$

(34)

which play the role of the 2-D wavelets.

A 2-D signal, i.e. an image, $f(x_1,x_2 )\in L^{2}(R^{2})$ can be projected into an $f_j (x_1,x_2)$ in $V_j^2 =V_j \otimes V_j$ by,

$$\begin{aligned} f_j (x_1,x_2 )=\sum _{n_1,n_2 =-\infty }^\infty {\left\langle {f(x_1,x_2 ),\phi _{j,(n_1,n_2 )}^2 } \right\rangle } \phi _{j,(n_1,n_2 )}^2. \end{aligned}$$

(35)

Similar to the 1-D case $f_j (x_1,x_2 )$ is interpreted as the approximation of $f$ at scale $2^{j}\hbox {d}x\times 2^{j}\hbox {d}y$, which is reconstructed from its samples $\langle {f(x_1,x_2 ),\phi _{j,(n_1,n_2 )}^2 } \rangle$ taken at the same scale. In addition, similar to the 1-D case, approximations at higher scales (lower resolutions) are obtained by removing details along the horizontal, vertical and diagonal directions of the image. The details removed across the vertical, horizontal and diagonal directions are given by the inner products $\langle {f(x_1,x_2 ),\psi _{n_1 ,n_2 }^{V_j \otimes W_j } (x_1,x_2 )}\rangle , \langle {f(x_1,x_2 ),\psi _{n_1,n_2 }^{W_j \otimes V_j } (x_1,x_2 )} \rangle$ and $\langle {f(x_1,x_2 ),\psi _{n_1,n_2 }^{W_j \otimes W_j } (x_1,x_2 )} \rangle$ respectively.

According to Eqs. 32–34 three separate projections, $d^{V_j \otimes W_j }, d^{W_j \otimes V_j }, d^{W_j \otimes W_j}$ in the corresponding orthogonal spaces $V_j \otimes W_j, W_j \otimes V_j, W_j \otimes W_j$ can be obtained. The projection in $V_j \otimes W_j$ is obtained by,

$$\begin{aligned} d^{V_j \otimes W_j }(x_1,x_2 )=\sum _{n_1,n_2 =-\infty }^\infty {\left\langle {f(x_1,x_2 ),\psi _{n_1,n_2 }^{V_j \otimes W_j } (x_1,x_2 )} \right\rangle } \psi _{n_1,n_2 }^{V_j \otimes W_j } (x_1,x_2 ) \end{aligned}$$

(36)

and the projections in $W_j \otimes V_j and W_j \otimes W_j$ are obtained by using similar equations.