## 1. Introduction

State-of-the-art mesoscale atmospheric models incorporate urban areas with physics-based parameterizations of processes that are not explicitly resolved at model grid scales (Chen et al. 2011). Current schemes partition urban-classified model grid cells, or mosaic urban “tiles,” into urban and nonurban contributions that are computed separately (Kusaka and Kimura 2004a; Li et al. 2013). The urban parameterization term accounting for development density, or urban fraction *f*_{urb} (note that the variables used in this paper are collected and defined in the appendix), is of primary importance to urban modeling since it defines the ratio of urban and nonurban contributions when aggregating the fields (i.e., sensible and latent heat fluxes, upward longwave radiation flux, albedo, and emissivity) to the model grid (Loridan and Grimmond 2012; Chen et al. 2011; Loridan et al. 2010). The urban contribution accounts for buildings and roads (e.g., Kusaka and Kimura 2004a; Grimmond et al. 2010), and the nonurban component, termed natural, is by default “*grassland*” to represent, for example, grass-covered urban parks (Kusaka and Kimura 2004a; Li et al. 2013). Loridan and Grimmond (2012) recommended that default parameters have higher values of *f*_{urb} together with a default natural class that has increased evaporation (“*cropland/natural vegetation mosaic*”). The role of *f*_{urb} was shown in Shaffer et al. (2015) to influence bias of various land–atmosphere interaction and surface energy balance terms within an arid city such as Phoenix, Arizona. For example, lower *f*_{urb} causes an overestimation of the latent heat flux. In addition, there may be multiple urban and nonurban land-cover types within an aggregated urban grid cell, suggesting that a mosaic approach may be of benefit when applied to urban areas (Li et al. 2013).

Ching (2013) summarized issues with state-of-the-art urban modeling and recommended that further guidance was needed when determining and aggregating development density from subgrid-resolution data. In this paper, we develop a method of parameterizing urban fraction for heterogeneous mosaic representations along with a means of assessing the representativeness of this parameterization. Monaghan et al. (2014) recently studied the influence of homogeneous development density versus heterogeneous development density

A method is proposed to assess probability density functions (PDF) of subgrid development density along with suggesting the use of the PDF’s mode to determine the spatially aggregated *h*. Shannon introduced the notion of entropy to measure the density of information (Billingsley 1965), and the concept was extended by Kolmogorov and Sinai for general nonlinear dynamical systems (Cornfeld et al. 1981). The normalized Shannon entropy provides a quantitative approach for evaluating the partitioning and representation of mosaic land-cover input data. The proposed method also provides guidance from an input-data perspective for multiscale and spatial evaluation of categorical partitioning schemes and motivates the use of mosaic methods, instead of a dominant-class approach, to accurately represent subgrid heterogeneity.

## 2. Methods

### a. Finescale urban land-cover data

*C*), which is composed of categorical data (Fry et al. 2011), and the percent-developed-imperviousness product (hereinafter Ψ), which is composed of continuous data (Xian et al. 2011). NLCD partitions the urban environment into four urban land-cover classes (

*C*

^{urb}, with

*C*

^{urb}⊂

*C*)—as developed open space (DOS), developed low intensity (DLI), developed medium intensity (DMI), and developed high intensity (DHI)—on the basis of thresholds of Ψ (Homer et al. 2004):

### b. Assessment of subgrid land-cover entropy

*γ*), treated as a random variable, with

*γ*as the fine grid, and Γ as the aggregated grid. From the PDF

*γ*within Γ

_{i,j}, at all grid points in Γ denoted

*p*(Ψ), and with Ψ having

*N*

_{s}states (i.e., with Ψ ∈ [1, 2, …, 100],

*N*

_{s}= 100), normalized Shannon entropy is defined as

*h*depends explicitly upon Γ, with Δ

_{Γ}and positioning of Γ with respect to

*γ*as implicit parameters that will influence the partitioning and aggregation of Ψ. Also,

*h*∈ [0, 1], where small

*h*(Γ

_{i,j}) indicate that

*p*

_{i,j}, denoted

*μ*

_{i,j}(or simply

*μ*), whereas

*h*

_{i,j}= 1 indicates that each state is equally probable. With additional categorical observations

*C*

^{urb}(

*γ*), we similarly construct the conditional PDF

*p*(Ψ |

*C*

^{urb}), for each

*C*

^{urb}, along with a conditional

*h*,

### c. Urban fraction parameter

*f*

_{urb}. In current applications, the total aggregate variable

*V*

_{total}for grid cells with urban contribution

*f*

_{urb}> 0 is produced for a variable

*V*(e.g., sensible heat flux) by a convex combination:

*V*

_{urban}is computed by an urban scheme for a particular

*C*

^{urb}and

*V*

_{nonurban}is computed by the LSM. The

*V*

_{nonurban}contribution is set by the natural class, which is typically set to

*cropland/natural vegetation mosaic*or

*grassland*by default as within the original implementation (Kusaka and Kimura 2004b; Chen et al. 2011; Li et al. 2013). An approach such as Eq. (4) precludes interactions between elements (trees shading buildings, etc.). Equation (4) suggests a linear sensitivity to the value of

*f*

_{urb}. Thus, model predictions will depend upon the selection of

*f*

_{urb}.

### d. Dominant subgrid approach to spatially heterogeneous f_{urb}

*α*of each

*C*

^{urb}within an aggregated grid cell Γ:

*C*

^{urb}based upon

*α*within each Γ for use within Eq. (4). For instance, the NUDAPT approach for obtaining

*C*

^{urb}data employs the weights

*w*(

*C*

^{urb}) = (0.5, 0.5, 0.9, 0.95). These weights represent the impervious fraction of an urban class, which is represented within the WRF urban schemes as buildings and roads. The WRF urban framework then converts these four NLCD urban classes into categories of low-intensity residential (LIR), high-intensity residential (HIR), and commercial/industrial/transportation (CIT; Ching 2013; Anderson et al. 1976), after identifying LIR = DOS ∪ DLI, HIR = DMI, and CIT = DHI (Glotfelty et al. 2013). Other urban-scheme parameters are derived (estimated) from additional data sources (Burian and Ching 2009; Glotfelty et al. 2013). We suggest an alternate approach for determining

*C*

^{urb}from the conditional PDF of subgrid observations such as with the NLCD Ψ:

*p*(Ψ |

*C*

^{urb}), denoted

### e. Mosaic subgrid approach to spatially heterogeneous f_{urb}

*C*

^{urb}. The present mosaic subgrid approach within WRF (Li et al. 2013) considers

*N*

_{t}homogeneous noninteracting subgrid tiles and aggregates contributions—for example—for a variable

*V*, based upon normalized (for

*N*

_{t}) tile areal fraction

*α*with class

*C*, by

*α*, and therefore

*N*

_{t}= 1 reproduces the dominant case. When

*c*∈

*C*

^{urb}, Eq. (4) is employed with

*f*

_{urb}= 0.5, 0.9, or 0.95 for LIR, HIR, or CIT, respectively. Again,

*V*

_{nonurban}is obtained from the LSM for a natural class, and for

*C*

^{urb}from NLCD the nonurban class (or impervious-cover contribution from Ψ) is not available.

### f. Verification experiments

A series of numerical experiments were conducted with the Advanced Research WRF Model to test the dominant and mosaic approaches to _{Γ} reduced by a factor of 3, such that the innermost nest *D*_{4} had Δ_{Γ} = 1 km. Domain *D*_{4} contained the entire Phoenix metropolitan area (PMA) within the interior of the domain to avoid lateral boundary issues (e.g., Warner et al. 1997). The first three domains were run with concurrent one-way nesting during premonsoon summer 2012 for the 3-day period beginning at 1800 UTC 17 June 2012 [see the Shaffer et al. (2015) Yonsei University–MM5 cases using modified morphological and material values for Phoenix (PHX-A/B) and their section 2b for details], with the outer domain initialized with NCEP Final Analysis (FNL; NCEP 1999) 6-hourly data. The 5-min-history archive interval on *D*_{3} was employed to provide consistent initial and lateral boundary forcing for the current set of experiments. We show several cases that test variations only for the innermost domain *D*_{4}. The Noah LSM (Chen and Dudhia 2001) was employed for parameterizing the land surface of nonurban classes, with the Single Layer Urban Canopy Model (SLUCM; Kusaka and Kimura 2004a) scheme for urban classes, within either the dominant or mosaic-tiling approaches.

A subset of the 2006 NLCD *C*^{urb} and Ψ datasets (described in section 2a) were obtained such that they contained the entire PMA. Since the NLCD *C*^{urb} data do not retain classification for nonurban contributions of urban-classified cells, default natural settings (*cropland/natural vegetation mosaic*) were employed. Nonurban *C* were obtained via Moderate Resolution Imaging Spectroradiometer (MODIS) 20-category 30-arc-s data modified for the Noah LSM, as discussed in Shaffer et al. (2015). For all simulations, urban areas outside the NLCD data subset were reclassified as *open shrubland*, the predominant nonurban *C* (within MODIS). Dominant *C*^{urb} was determined by maximum *α* for the NLCD *C*^{urb} dataset excluding DOS. To avoid parameter tuning, the default urban parameter values (invariant for all cases) are used except for *f*_{urb}, as described below.

The cases tested for *f*_{urb}. A case was tested with *N*_{t} = 3. To simplify the demonstration of the proposed methods, only the dominant *μ*, *w*_{mos} cases will be shown in this paper.

### g. Verification with observations

Observations of near-surface air temperature were obtained with the West Phoenix Flux Tower [WPHX-FT; described in Chow et al. (2014) and equipped with Vaisala, Inc., model HMP45AC temperature–relative humidity sensors within a radiation shield, with 1-Hz sampling], along with micrometeorological stations that are deployed within the Flood Control District of Maricopa County (FCDMC) Automated Local Evaluation in Real Time system (hereinafter ALERT; data were obtained at http://www.fcd.maricopa.gov/Weather/weather.aspx). The ALERT stations use either Vaisala HUMICAP model HMP155 humidity and temperature probes within a radiation shield or Hydrolinx Systems, Inc., model 2048RH/T relative humidity and temperature sensors (as indicated in Table 2), with 15-min sampling. These data were averaged to 30-min intervals, along with 5-min instantaneous WRF output for the grid cell containing the station.

Summary of station metadata. WPHX-FT is the only station not from the FCDMC ALERT system. Temperature-sensor types are indicated with a superscript in the sensor-identifier (ID) column: V is Vaisala HMP155 and H is Hydrolinx 2048RH/T. Note that the XRD site elevation slopes down to a catch basin east of the station.

A subset of 11 ALERT stations was selected for analysis on the basis of the criteria of being within the PMA study area and within modified urban-classified WRF grid cells. Basic station metadata are summarized in Table 2. No corrections were applied for sensor height, in comparison with the WRF 2-m diagnostic temperature *T*_{2m}. Standard statistical measures (Willmott 1981; Willmott et al. 1985) employed were ordinary least squares regression between observed and predicted values, mean bias error (MBE), and mean absolute error (MAE), along with the modified index of agreement for MAE *d*_{1} (a dimensionless statistical measure of relative average error), root-mean-square error (RMSE), systematic (linear model bias) and unsystematic (model precision) RMSE (RMSE_{s} and RMSE_{u}, respectively, where RMSE^{2} = *d*_{2}. Comparisons were made with the composite diurnal variation for the 3-day study period at each station and across all stations (ALERT and WPHX-FT), for each simulation case.

## 3. Results and discussion

### a. Multiscale analysis of input data

The NLCD 2006 data described in section 2a for Ψ and *C* are presented in Figs. 1a and 1b for a particular 9 km × 9 km subset of the PMA containing the WPHX-FT (Chow et al. 2014). The probability distribution of Ψ is assessed for each *C*^{urb} (Figs. 1c,d). For this particular PMA subset, *μ* = 57% with a largest-area *C*^{urb} of DMI. Conditional PDFs, or *p*(Ψ | *C*^{urb}), are shown for each *C*^{urb} (Fig. 1d) for the data in Figs. 1a and 1b. Apparent in Figs. 1c and 1d is that the limits of Ψ for each *C*^{urb} [Eq. (1)] are not strictly valid. There may be misclassification or other differences between these data, or processing differences between the two NLCD datasets (Ψ and *C*^{urb}). In the analysis that is presented here, no modifications are made to account for these inherent NLCD data discrepancies.

The merging of DOS with DLI to form LIR (e.g., Figs. 1c,d) overestimates both the area and development density and changes the *C*^{urb} partition [Eq. (1)]. This result indicates that urbanization within *C*^{urb} Eq. (5). Furthermore, the NUDAPT approach, with *w*(DOS ∪ DLI) = 0.5, additionally overrepresents the density in addition to *α*. Given the many parameters in the urban models (e.g., Grimmond et al. 2011), deficiencies in

The multiscale influence of horizontal aggregate length Δ_{Γ} is examined in Fig. 2 with center Γ_{0} at the WPHX-FT. For these analyses, Δ_{Γ} varies from 30 to 9990 m. The roles of Δ_{Γ} on *h* [Eq. (2)] for *p*(Ψ) and on *p*(Ψ | *C*^{urb}) are shown (Fig. 2a) for the four *C*^{urb}. For this aggregated gridcell center, the DOS have low _{Γ} < 600 m, owing to the low number of DOS-classified fine-grid cells within this Δ_{Γ} range. The *C*^{urb} seem to converge to distinct values, principally dependent upon *C*^{urb} for Δ_{Γ} > 1 × 10^{3}–3 × 10^{3} m, and upon partition [Eq. (1)] of Ψ. This large Δ_{Γ} behavior suggests that there may exist a “citywide” maximum entropy for *p*(Ψ | *C*^{urb}). The increase of _{Γ} also indicates that the *p*(Ψ) by *C*^{urb}, as *p*(Ψ | *C*^{urb}), reduces *h*.

Multiscale evaluation with fixed Γ_{0} at WPHX-FT with *C*^{urb} and Ψ from NLCD 2006 data products for (a) *h* for *p*(Ψ) with Eq. (2) (black) and *p*(Ψ | *C*^{urb}) with Eq. (3) (colored by *C*^{urb}; bottom legend) and for (b) *f*_{urb} (top legend) from using just *C*^{urb} with Eq. (5) with *w*(*C*^{urb}) = (0.50, 0.50, 0.90, 0.95) (labeled NUDAPT), and following thresholds of Ψ as in Eq. (1), with *w*(*C*^{urb}) = (0.20, 0.50, 0.80, 1.0) (labeled Max Ψ) and for *w*(*C*^{urb}) = (0.20, 0.50, 0.80, 0.95) (labeled w/DOS). (c) The *f*_{urb} estimated by mean, median, and mode with Eq. (6), for just using *p*(Ψ) (labeled All Ψ) and *p*(Ψ | *C*^{urb}) (bottom legend). Thresholds of Ψ per Eq. (1) are indicated by the dash–dotted horizontal lines (colored by *C*^{urb}) in (b) and (c). The *C*^{urb} color scheme in (a) and (c) is the same as in Figs. 1a, 1c, and 1d. The _{Γ} > ~3 km), which are reduced for categorical partitioning, indicating improved parameterization of *f*_{urb}, which varies with Δ_{Γ}, along with skewness of *p*(Ψ | *C*^{urb}).

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

Multiscale evaluation with fixed Γ_{0} at WPHX-FT with *C*^{urb} and Ψ from NLCD 2006 data products for (a) *h* for *p*(Ψ) with Eq. (2) (black) and *p*(Ψ | *C*^{urb}) with Eq. (3) (colored by *C*^{urb}; bottom legend) and for (b) *f*_{urb} (top legend) from using just *C*^{urb} with Eq. (5) with *w*(*C*^{urb}) = (0.50, 0.50, 0.90, 0.95) (labeled NUDAPT), and following thresholds of Ψ as in Eq. (1), with *w*(*C*^{urb}) = (0.20, 0.50, 0.80, 1.0) (labeled Max Ψ) and for *w*(*C*^{urb}) = (0.20, 0.50, 0.80, 0.95) (labeled w/DOS). (c) The *f*_{urb} estimated by mean, median, and mode with Eq. (6), for just using *p*(Ψ) (labeled All Ψ) and *p*(Ψ | *C*^{urb}) (bottom legend). Thresholds of Ψ per Eq. (1) are indicated by the dash–dotted horizontal lines (colored by *C*^{urb}) in (b) and (c). The *C*^{urb} color scheme in (a) and (c) is the same as in Figs. 1a, 1c, and 1d. The _{Γ} > ~3 km), which are reduced for categorical partitioning, indicating improved parameterization of *f*_{urb}, which varies with Δ_{Γ}, along with skewness of *p*(Ψ | *C*^{urb}).

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

Multiscale evaluation with fixed Γ_{0} at WPHX-FT with *C*^{urb} and Ψ from NLCD 2006 data products for (a) *h* for *p*(Ψ) with Eq. (2) (black) and *p*(Ψ | *C*^{urb}) with Eq. (3) (colored by *C*^{urb}; bottom legend) and for (b) *f*_{urb} (top legend) from using just *C*^{urb} with Eq. (5) with *w*(*C*^{urb}) = (0.50, 0.50, 0.90, 0.95) (labeled NUDAPT), and following thresholds of Ψ as in Eq. (1), with *w*(*C*^{urb}) = (0.20, 0.50, 0.80, 1.0) (labeled Max Ψ) and for *w*(*C*^{urb}) = (0.20, 0.50, 0.80, 0.95) (labeled w/DOS). (c) The *f*_{urb} estimated by mean, median, and mode with Eq. (6), for just using *p*(Ψ) (labeled All Ψ) and *p*(Ψ | *C*^{urb}) (bottom legend). Thresholds of Ψ per Eq. (1) are indicated by the dash–dotted horizontal lines (colored by *C*^{urb}) in (b) and (c). The *C*^{urb} color scheme in (a) and (c) is the same as in Figs. 1a, 1c, and 1d. The _{Γ} > ~3 km), which are reduced for categorical partitioning, indicating improved parameterization of *f*_{urb}, which varies with Δ_{Γ}, along with skewness of *p*(Ψ | *C*^{urb}).

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

The multiscale role of Δ_{Γ} on derived *f*_{urb} is presented (Fig. 2b) for the corresponding distributions with *h* (Fig. 2a) that are discussed above. Shown are derived *C*^{urb} following Eq. (5) with *w*(*C*^{urb}) = (0.50, 0.50, 0.90, 0.95) (labeled as NUDAPT), for following thresholds of Ψ as in Eq. (1) with *w*(*C*^{urb}) = (0.20, 0.50, 0.80, 1.0) (labeled as Max Ψ), and for *w*(*C*^{urb}) = (0.20, 0.50, 0.80, 0.95) (labeled as w/DOS). Also shown are *f*_{urb} estimated by mean, median, and mode [Eq. (6)], for just using *p*(Ψ) [labeled as *p*(Ψ)] and *p*(Ψ | *C*^{urb}) (labeled by *C*^{urb}). All *f*_{urb} are Δ_{Γ} dependent. The NUDAPT *f*_{urb} values are ~0.1% above those obtained with limits given in Eq. (1), which differ by less than 2% for Δ_{Γ} of less than approximately 3 km (Fig. 2b). The w/DOS values reduce from ~84% below 1 km to ~80% for Δ_{Γ} of less than approximately 3 km and then decrease to ~73% for Δ_{Γ} ≈ 9 km. The w/DOS *f*_{urb} are 0.02%–0.15% above the mean and median *p*(Ψ) method by Eq. (5) but are 0.1%–0.2% below *μ*[*p*(Ψ)] for Δ_{Γ} > ~1 km, which varies between *μ*_{DMI} and *μ*_{DHI} as Γ increases. For this Γ_{0}, the *C*^{urb} of DOS has *f*_{urb} values decrease with Δ_{Γ} > 500 m for the *p*(Ψ) single-class method but remain bounded [by Eq. (1)] with skewed distribution (for this Γ_{0}) for *p*(Ψ | *C*^{urb}) mosaic methods.

### b. Spatial analysis of input data

To examine the spatial distribution of *h* values so as to assess *C*^{urb}, this analysis will focus on a fixed Δ_{Γ} (990 m), which is comparable to the typical finescale (1 km) of mesoscale simulation studies. The approach using a single dominant *C*^{urb} [Eq. (5)] is presented for the region containing the PMA (Fig. 3). An estimate of *μ* of *p*(Ψ) is shown as a map in Fig. 3a. The urban-core area is present in Fig. 3a, but fringe areas that incorporate DOS with low Ψ values are absent. The corresponding *h* (Fig. 3b) indicates that *p*(Ψ) is near *μ* outside the urban core, but *p*(Ψ) are less represented by *μ* within the urban-core areas. These large *h* values indicate increased heterogeneity of Ψ with urban-core areas and motivate investigating mosaic approaches.

Maps of derived _{Γ} = 990 m for the PMA by (a) mode of *p*(Ψ) and by Eq. (5) for (c) *w*(*C*^{urb}) = (0.50, 0.50, 0.90, 0.95) NUDAPT method and (d) *w*(*C*^{urb}) = (0.20, 0.50, 0.80, 0.95) NUDAPT method following Eq. (1). (b) The *h* (dimensionless) for *p*(Ψ) corresponding to the *μ* for *p*(Ψ) alone as in (a) underestimates the extent of the urban *C* and has large *h* for much of the urban area as in (b). The NUDAPT approach overestimates the urban area and *f*_{urb} when combining DOS with DLI [(c) vs (d)].

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

Maps of derived _{Γ} = 990 m for the PMA by (a) mode of *p*(Ψ) and by Eq. (5) for (c) *w*(*C*^{urb}) = (0.50, 0.50, 0.90, 0.95) NUDAPT method and (d) *w*(*C*^{urb}) = (0.20, 0.50, 0.80, 0.95) NUDAPT method following Eq. (1). (b) The *h* (dimensionless) for *p*(Ψ) corresponding to the *μ* for *p*(Ψ) alone as in (a) underestimates the extent of the urban *C* and has large *h* for much of the urban area as in (b). The NUDAPT approach overestimates the urban area and *f*_{urb} when combining DOS with DLI [(c) vs (d)].

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

Maps of derived _{Γ} = 990 m for the PMA by (a) mode of *p*(Ψ) and by Eq. (5) for (c) *w*(*C*^{urb}) = (0.50, 0.50, 0.90, 0.95) NUDAPT method and (d) *w*(*C*^{urb}) = (0.20, 0.50, 0.80, 0.95) NUDAPT method following Eq. (1). (b) The *h* (dimensionless) for *p*(Ψ) corresponding to the *μ* for *p*(Ψ) alone as in (a) underestimates the extent of the urban *C* and has large *h* for much of the urban area as in (b). The NUDAPT approach overestimates the urban area and *f*_{urb} when combining DOS with DLI [(c) vs (d)].

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

The method of a single dominant *C*^{urb} in Fig. 3a for *μ* of *p*(Ψ) underestimates the extent of the urban area relative to the NUDAPT approach (Fig. 3c) for weights *w*(*C*^{urb}) = (0.50, 0.50, 0.90, 0.95) in Eq. (5). A modified approach satisfying Eq. (1) with *w*(*C*^{urb}) = (0.20, 0.50, 0.80, 0.95) is presented in Fig. 3d. As with Γ_{0} = WPHX-FT (Fig. 1 and Fig. 2), the NUDAPT approach of combining DOS and DLI overestimates the areal extent of LIR (Fig. 3c), along with *w*(*C*^{urb}) overestimating

Analysis at Δ_{Γ} = 990 m of *C*^{urb} being DOS (Figs. 4a and 5a), DLI (Figs. 4b and 5b), DMI (Figs. 4c and 5c), and DHI (Figs. 4d and 5d). The *C*^{urb} as per Eq. (1), with tile aggregation using the respective subgrid areal fractions as per, for example, Eq. (7). Heterogeneous structure is present within *C*^{urb}, with *p*(Ψ | *C*^{urb}) are less represented by

Similar to Fig. 3, but showing a map of *C*^{urb} being (a) DOS, (b) DLI, (c) DMI, and (d) DHI. The spatial pattern of most probable Ψ for each *C*^{urb} shows where each class contributes to the aggregated model output [Eq. (7)]. Note that *α*(Γ | *C*^{urb}) can be similarly displayed, with the product *C*^{urb} in Eq. (7).

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

Similar to Fig. 3, but showing a map of *C*^{urb} being (a) DOS, (b) DLI, (c) DMI, and (d) DHI. The spatial pattern of most probable Ψ for each *C*^{urb} shows where each class contributes to the aggregated model output [Eq. (7)]. Note that *α*(Γ | *C*^{urb}) can be similarly displayed, with the product *C*^{urb} in Eq. (7).

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

Similar to Fig. 3, but showing a map of *C*^{urb} being (a) DOS, (b) DLI, (c) DMI, and (d) DHI. The spatial pattern of most probable Ψ for each *C*^{urb} shows where each class contributes to the aggregated model output [Eq. (7)]. Note that *α*(Γ | *C*^{urb}) can be similarly displayed, with the product *C*^{urb} in Eq. (7).

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

As in Fig. 4, but following Eq. (3) for *C*^{urb} being (a) DOS, (b) DLI, (c) DMI, and (d) DHI. Spatial guidance for categorical partition [Eq. (1)] and parameterization of *f*_{urb} [Eq. (6)] indicates that DOS and DHI have lower *p*(Ψ | *C*^{urb}) and where the parameterization [Eq. (6)] may be less representative of the subgrid development density distribution or an alternate partitioning scheme to Eq. (1) may be needed.

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

As in Fig. 4, but following Eq. (3) for *C*^{urb} being (a) DOS, (b) DLI, (c) DMI, and (d) DHI. Spatial guidance for categorical partition [Eq. (1)] and parameterization of *f*_{urb} [Eq. (6)] indicates that DOS and DHI have lower *p*(Ψ | *C*^{urb}) and where the parameterization [Eq. (6)] may be less representative of the subgrid development density distribution or an alternate partitioning scheme to Eq. (1) may be needed.

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

As in Fig. 4, but following Eq. (3) for *C*^{urb} being (a) DOS, (b) DLI, (c) DMI, and (d) DHI. Spatial guidance for categorical partition [Eq. (1)] and parameterization of *f*_{urb} [Eq. (6)] indicates that DOS and DHI have lower *p*(Ψ | *C*^{urb}) and where the parameterization [Eq. (6)] may be less representative of the subgrid development density distribution or an alternate partitioning scheme to Eq. (1) may be needed.

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

### c. Evaluation of method within WRF

Evaluation with many stations across the PMA, where station locations, dominant *C*, and terrain elevation are shown in Fig. 6, with details given in Table 2, was done to assess the impact on a typically observed meteorological variable for the methods of determining a spatially heterogeneous *f*_{urb}. Figure 7 and Fig. 8 show diurnal variation (in 30-min intervals averaged over the 3-day study period) of the near-ground air temperature *T*_{2m} for these stations, for the Max Ψ, *w*_{mos} cases. Diurnal variations of statistical measures across all stations are shown for these model cases in Fig. 9, and overall statistical measures across all stations for the entire study period are presented in Table 3 for the model cases given in Table 1.

Maps of WPHX-FT and ALERT station locations (black circles; yellow circles are within grid cells with *f*_{urb} > 0 and are used for analysis, with station ID string as in Table 2) within the Phoenix metropolitan study area, along with (a) dominant land-cover class (described in text) and (b) terrain elevation (color bar). Also shown in (b) is the boundary of grid cells with

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

Maps of WPHX-FT and ALERT station locations (black circles; yellow circles are within grid cells with *f*_{urb} > 0 and are used for analysis, with station ID string as in Table 2) within the Phoenix metropolitan study area, along with (a) dominant land-cover class (described in text) and (b) terrain elevation (color bar). Also shown in (b) is the boundary of grid cells with

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

Maps of WPHX-FT and ALERT station locations (black circles; yellow circles are within grid cells with *f*_{urb} > 0 and are used for analysis, with station ID string as in Table 2) within the Phoenix metropolitan study area, along with (a) dominant land-cover class (described in text) and (b) terrain elevation (color bar). Also shown in (b) is the boundary of grid cells with

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

Diurnal 30-min *T*_{2m} during 17–20 Jun 2012 for *μ* and *α* from Ψ > 0 (green squares), with *α* from max *α*_{c} (red times signs), and for

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

Diurnal 30-min *T*_{2m} during 17–20 Jun 2012 for *μ* and *α* from Ψ > 0 (green squares), with *α* from max *α*_{c} (red times signs), and for

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

Diurnal 30-min *T*_{2m} during 17–20 Jun 2012 for *μ* and *α* from Ψ > 0 (green squares), with *α* from max *α*_{c} (red times signs), and for

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

As in Fig. 7, but for ALERT stations (a) MKN, (b) O64, (c) P2B, (d) PJX, (e) WBG, and (f) XRD.

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

As in Fig. 7, but for ALERT stations (a) MKN, (b) O64, (c) P2B, (d) PJX, (e) WBG, and (f) XRD.

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

As in Fig. 7, but for ALERT stations (a) MKN, (b) O64, (c) P2B, (d) PJX, (e) WBG, and (f) XRD.

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

Diurnal plots of statistical measures (Willmott 1981; Willmott et al. 1985) for each simulation case (Table 1) over all 12 stations (Table 2) for near-ground air temperature: (a) MAE, (b) index of agreement for RMSE *d*_{2}, (c) MBE, (d) RMSE, (e) systematic error RMSE_{s}, and (f) unsystematic error RMSE_{u}. See Table 3 for daily totals. The *d*_{1} statistic showed a ranking of models that was similar to that for *d*_{2} (but with lower values) and so is not shown.

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

Diurnal plots of statistical measures (Willmott 1981; Willmott et al. 1985) for each simulation case (Table 1) over all 12 stations (Table 2) for near-ground air temperature: (a) MAE, (b) index of agreement for RMSE *d*_{2}, (c) MBE, (d) RMSE, (e) systematic error RMSE_{s}, and (f) unsystematic error RMSE_{u}. See Table 3 for daily totals. The *d*_{1} statistic showed a ranking of models that was similar to that for *d*_{2} (but with lower values) and so is not shown.

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

Diurnal plots of statistical measures (Willmott 1981; Willmott et al. 1985) for each simulation case (Table 1) over all 12 stations (Table 2) for near-ground air temperature: (a) MAE, (b) index of agreement for RMSE *d*_{2}, (c) MBE, (d) RMSE, (e) systematic error RMSE_{s}, and (f) unsystematic error RMSE_{u}. See Table 3 for daily totals. The *d*_{1} statistic showed a ranking of models that was similar to that for *d*_{2} (but with lower values) and so is not shown.

Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1

Statistical measures (Willmott 1981; Willmott et al. 1985) for each simulation case (Table 1) over all 12 stations (Table 2) for the 3-day study period for near-ground air temperature. Here, *a*, *b*, and *r*^{2} are the intercept, slope, and correlation coefficient, respectively, for an ordinary least squares regression between observed and predicted values. For simulation vs observations, MBE and MAE are the mean bias and absolute errors and *d*_{1} is the modified index of agreement for MAE. Also given are the systematic, unsystematic, and total RMSE and the index of agreement *d*_{2}.

Figure 7 shows that *T*_{2m} is often underestimated for Max Ψ and *w*_{mos} case shows improvement at most stations (Figs. 7, 8), with improved overall statistical measures during the entire day (Fig. 9) with the exception of midafternoon, for which some measures (e.g., RMSE_{s}; Fig. 9e) degrade between 1030 and 1630 LST. Regardless, the overall performance of *w*_{mos} is comparable to mosaic (Table 3), and both are correspondingly much better than the dominant methods.

Accounting for the stated thresholds (Max Ψ) given by Eq. (1) shows a decrease in performance, which is attributable to the relative contributions of Noah and SLUCM [Eq. (4)]. We note that the default WRF SLUCM model parameters were recommended by Loridan and Grimmond (2012) on the basis of optimization [following Loridan et al. (2010)] of surface energy balance terms (net radiation *Q*_{*}, sensible heat flux *Q*_{H}, and latent heat flux *Q*_{E}) for many cities and that because we here modify the *f*_{urb} values, together with the parameters not being adjusted for Phoenix (e.g., Shaffer et al. 2015) or specifically for each station, such a decrease in model performance may be anticipated.

We hypothesize that the fact that the mosaic cases for the proposed modified scheme do not show improved performance over the default mosaic (not shown) is likely due to a combined effect of both urban-class parameter values (we simply employed default values) and the default selection of the natural class. As discussed by Loridan and Grimmond (2012), there are several factors involved with the derivation of default parameters, where higher *f*_{urb} values were selected together with a natural class with increased evaporation (*cropland/natural vegetation mosaic*), which provided better agreement with their observations. Lower values of *f*_{urb} were shown by Shaffer et al. (2015) to cause an overestimation of the latent heat flux within arid cities such as Phoenix. Here again, the natural class is uniformly set to the default class rather than being derived from observations of *C* and is expected to introduce such a bias in latent heat flux, and in turn, the near-ground air temperature, as observed for the selected meteorological stations. We note that some stations were near grass (parks) or croplands while others were in xeric vegetation settings (open shrubland). No correction for station-setting bias was employed here.

The widely used NLCD *C* unfortunately does not retain nonurban-class contribution within urban-classified grid cells. Furthermore, other NLCD classes are categorically the majority at Δ_{γ}. Thus, higher-resolution categorical observations, such as the National Agricultural Imagery Program (NAIP), which has Δ_{γ} = 1 m for the PMA (Li et al. 2014), would enable deriving percent contributions of these nonurban classes and would allow for determining gridded values of majority natural class within each urban tile. Such an approach should reduce bias for both default and modified mosaic approaches. As discussed previously regarding the input data, the weights of 0.5, 0.9, and 0.95 for the default mosaic are actually overestimating the actual development density (or plan areal fraction of built environment). The fairly robust evaluation of default mosaic may result from compensation from bias of other urban parameter values and requires further evaluation, which is beyond the scope of this paper (e.g., see Loridan and Grimmond 2012). Generalization of urban class with more gridded parameters and for more classifications or for classification schemes that differ from Eq. (1) should also be explored.

## 4. Conclusions

Incorporating spatially heterogeneous development density allows for improvement of urban parameterization, including for methods that use a single dominant *C*^{urb}. A more accurate representation of *f*_{urb} [e.g., Eqs. (4) and (7)]. Furthermore, *h* and indicates a more representative parameterization of *f*_{urb}. Employing data products (e.g., NLCD) derived from satellite observations, in principle, enables deriving *p*(Ψ) and *p*(Ψ | *C*^{urb}) motivate consideration of the subgrid PDF and determining alternative methods of representing subgrid flux contributions, such as the most probable value. Employing a categorical partitioning scheme and conditional PDFs enable determining

The normalized Shannon entropy provides a quantitative means for assessing the representation of subgrid PDFs of development density. Spatial analysis of *h* is useful for determining where distributions of subgrid development density are not well represented by the mode of a PDF and also for determining where a categorical partitioning scheme may be misrepresenting urban heterogeneity. Assessing *h* also provides important guidance for determining possible sources of bias (e.g., of *h* values indicate increased subgrid heterogeneity, which may result in reduced confidence for the parameterization (*C*^{urb}. The increase of _{Γ} also indicates that the *p*(Ψ) by *C*^{urb}, as *p*(Ψ | *C*^{urb}), with a reduced *h*.

These analyses of input data motivate employing a mosaic urban approach, along with investigating categorical partitioning schemes, informed by conditional *h*, which enable one to assess the partitioning of urban tiles at each model grid cell. Additional classes are sensible provided they are physically based and will guide parameter selection within urban models. These results also motivate investigating alternate partitioning schemes to categorically segregate *C* for model physical parameters, the spatial variations of which are not easily derived from remote observations. Supplementary datasets may enable refinement of the process of estimating, for example, material property parameters, particularly albedo, as was done here with Ψ for *C* products together are not sufficient to determine aggregated nonurban *C* since any NLCD grid cell with Ψ > 0 will be classified as urban. Employing higher-resolution *C* data (e.g., NAIP; Li et al. 2014) would enable the construction of a fractional contribution for each *C* at various aggregated scales, rather than attributing *grassland* to the nonurban *C*.

Multiscale analyses of *h* indicate citywide maximum values at large scales (>~3 km), which are also dependent upon class partitioning. Note that citywide Ψ distributions also depend upon the city, as shown in Zhang et al. (2012). The _{Γ} and Γ_{0}. No significant variation of model prediction across scales from 9 km to 333 m was found by Shaffer et al. (2015), owing to homogeneous scale-independent values of *f*_{urb}. Employing a resolution-dependent *f*_{urb} that is based upon improved parameterization of input data provides one means of influencing simulation prediction at various aggregated grid scales, which is of particular importance for models with variable grid size within a domain (e.g., Skamarock et al. 2012).

The multiscale methods presented herein can be applied for more general numerical prediction models for mixed land use and land cover, such as within urban environments. Here we tested with WRF employing Noah with SLUCM for dominant and mosaic approaches with the NLCD datasets. These selections were made to address an important weakness in the current approach of deriving grid-scale urban fraction from 30-m NLCD data in the widely used WRF-Urban model. Alternate models, datasets, class-partitioning schemes, and flux-aggregation approaches could be similarly examined with these methods.

## Acknowledgments

This work was supported by grants awarded to Arizona State University (ASU) from the National Science Foundation (NSF) under Grant DMS 1419593, U.S. Department of Agriculture and National Institute of Food and Agriculture (USDA-NIFA) Grant 2015-67003-23508, and NSF Grants EF 1049251 and EAR 1204774. We acknowledge high-performance computing support from Yellowstone (ark:/85065/d7wd3xhc) provided by the National Center for Atmospheric Research’s Computational and Information Systems Laboratory, sponsored by the NSF, along with support from ASU Research Computing. West Phoenix Flux Tower data are available from Central Arizona Phoenix Long-Term Ecological Research (site manager Phil Torrant) with funding provided by Grant CAP3: BCS-1026865 and by NSF via EaSM Grant EF-1049251. We thank Daniel Henz for supplying the Flood Control District of Maricopa County ALERT system station data (available at http://www.fcd.maricopa.gov/Weather/weather.aspx). We also thank the three anonymous reviewers for feedback that improved the paper.

## APPENDIX

### Summary of Nomenclature

C | Land-cover class |

C^{urb} | Urban land-cover class |

f_{urb} | Urban fraction parameter |

Heterogeneous development density | |

Heterogeneous development density for mosaic approach | |

h | Normalized Shannon entropy |

Conditional normalized Shannon entropy | |

N_{s} | Number of states (of |

N_{t} | Number of tiles for mosaic approach |

p() | Probability density function |

Q_{*} | Net radiation |

Q_{E} | Latent heat flux |

Q_{H} | Sensible heat flux |

V | Generic model variable |

V_{total} | Total aggregated |

V_{urban} | Urban contribution of |

V_{nonurban} | Nonurban contribution of |

w | Weighting coefficient |

α | Normalized area fraction |

Γ | Aggregated grid |

Γ_{0} | Center of aggregated grid cell |

γ | Fine grid |

Δ_{Γ} | Aggregated grid length scale |

Δ_{γ} | Fine-grid length scale |

μ | Mode of |

Mode of | |

Ψ | Percent developed imperviousness |

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