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 furb (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 furb together with a default natural class that has increased evaporation (“cropland/natural vegetation mosaic”). The role of furb 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 furb 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 
2. Methods
a. Finescale urban land-cover data
b. Assessment of subgrid land-cover entropy
c. Urban fraction parameter
d. Dominant subgrid approach to spatially heterogeneous furb
e. Mosaic subgrid approach to spatially heterogeneous furb
f. Verification experiments
A series of numerical experiments were conducted with the Advanced Research WRF Model to test the dominant and mosaic approaches to 
A subset of the 2006 NLCD Curb and Ψ datasets (described in section 2a) were obtained such that they contained the entire PMA. Since the NLCD Curb 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 Curb was determined by maximum α for the NLCD Curb dataset excluding DOS. To avoid parameter tuning, the default urban parameter values (invariant for all cases) are used except for furb, as described below.
The cases tested for 
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 T2m. 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 d1 (a dimensionless statistical measure of relative average error), root-mean-square error (RMSE), systematic (linear model bias) and unsystematic (model precision) RMSE (RMSEs and RMSEu, respectively, where RMSE2 = 
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 Curb (Figs. 1c,d). For this particular PMA subset, μ = 57% with a largest-area Curb of DMI. Conditional PDFs, or p(Ψ | Curb), are shown for each Curb (Fig. 1d) for the data in Figs. 1a and 1b. Apparent in Figs. 1c and 1d is that the limits of Ψ for each Curb [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 Curb). In the analysis that is presented here, no modifications are made to account for these inherent NLCD data discrepancies.
Analysis of NLCD 2006 data of a 9 km × 9 km subset of the PMA (containing the West Phoenix Flux Tower at 33.484°N, 112.143°W, labeled WPHX FT Subset and denoted by a solid black circle), for maps of the (a) categorical C and (b) continuous Ψ and for (c) the distribution of cumulative count of Ψ colored by C, with count per C and total count indicated, and (d) PDFs p(Ψ | Curb) for Curb = DOS, DLI, DMI, and DHI. The same C color scheme is used for (a),(c), and (d). Conditional PDFs are constructed by partitioning Ψ with C. Assessing distributions of Ψ for each C reveals bias of methods that use just one of these data products to derive furb and reveals furb parameterization bias for each C.
Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1
Analysis of NLCD 2006 data of a 9 km × 9 km subset of the PMA (containing the West Phoenix Flux Tower at 33.484°N, 112.143°W, labeled WPHX FT Subset and denoted by a solid black circle), for maps of the (a) categorical C and (b) continuous Ψ and for (c) the distribution of cumulative count of Ψ colored by C, with count per C and total count indicated, and (d) PDFs p(Ψ | Curb) for Curb = DOS, DLI, DMI, and DHI. The same C color scheme is used for (a),(c), and (d). Conditional PDFs are constructed by partitioning Ψ with C. Assessing distributions of Ψ for each C reveals bias of methods that use just one of these data products to derive furb and reveals furb parameterization bias for each C.
Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1
Analysis of NLCD 2006 data of a 9 km × 9 km subset of the PMA (containing the West Phoenix Flux Tower at 33.484°N, 112.143°W, labeled WPHX FT Subset and denoted by a solid black circle), for maps of the (a) categorical C and (b) continuous Ψ and for (c) the distribution of cumulative count of Ψ colored by C, with count per C and total count indicated, and (d) PDFs p(Ψ | Curb) for Curb = DOS, DLI, DMI, and DHI. The same C color scheme is used for (a),(c), and (d). Conditional PDFs are constructed by partitioning Ψ with C. Assessing distributions of Ψ for each C reveals bias of methods that use just one of these data products to derive furb and reveals furb parameterization bias for each C.
Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1
The merging of DOS with DLI to form LIR (e.g., Figs. 1c,d) overestimates both the area and development density and changes the Curb partition [Eq. (1)]. This result indicates that urbanization within 
The multiscale influence of horizontal aggregate length ΔΓ is examined in Fig. 2 with center Γ0 at the WPHX-FT. For these analyses, ΔΓ varies from 30 to 9990 m. The roles of ΔΓ on h [Eq. (2)] for p(Ψ) and on 
Multiscale evaluation with fixed Γ0 at WPHX-FT with Curb and Ψ from NLCD 2006 data products for (a) h for p(Ψ) with Eq. (2) (black) and 
Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1
Multiscale evaluation with fixed Γ0 at WPHX-FT with Curb and Ψ from NLCD 2006 data products for (a) h for p(Ψ) with Eq. (2) (black) and
Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1
Multiscale evaluation with fixed Γ0 at WPHX-FT with Curb and Ψ from NLCD 2006 data products for (a) h for p(Ψ) with Eq. (2) (black) and
Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1
The multiscale role of ΔΓ on derived furb is presented (Fig. 2b) for the corresponding distributions with h (Fig. 2a) that are discussed above. Shown are derived 
b. Spatial analysis of input data
To examine the spatial distribution of h values so as to assess 
Maps of derived 
Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1
Maps of derived
Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1
Maps of derived
Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1
The method of a single dominant Curb 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(Curb) = (0.50, 0.50, 0.90, 0.95) in Eq. (5). A modified approach satisfying Eq. (1) with w(Curb) = (0.20, 0.50, 0.80, 0.95) is presented in Fig. 3d. As with Γ0 = 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(Curb) overestimating 
Analysis at ΔΓ = 990 m of 
Similar to Fig. 3, but showing a map of 
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
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
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 
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
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
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 furb. 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 T2m for these stations, for the Max Ψ, 
Maps of WPHX-FT and ALERT station locations (black circles; yellow circles are within grid cells with furb > 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 furb > 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 furb > 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 T2m during 17–20 Jun 2012 for 
Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1
Diurnal 30-min T2m during 17–20 Jun 2012 for
Citation: Journal of Applied Meteorology and Climatology 55, 9; 10.1175/JAMC-D-16-0027.1
Diurnal 30-min T2m during 17–20 Jun 2012 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 d2, (c) MBE, (d) RMSE, (e) systematic error RMSEs, and (f) unsystematic error RMSEu. See Table 3 for daily totals. The d1 statistic showed a ranking of models that was similar to that for d2 (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 d2, (c) MBE, (d) RMSE, (e) systematic error RMSEs, and (f) unsystematic error RMSEu. See Table 3 for daily totals. The d1 statistic showed a ranking of models that was similar to that for d2 (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 d2, (c) MBE, (d) RMSE, (e) systematic error RMSEs, and (f) unsystematic error RMSEu. See Table 3 for daily totals. The d1 statistic showed a ranking of models that was similar to that for d2 (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 r2 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 d1 is the modified index of agreement for MAE. Also given are the systematic, unsystematic, and total RMSE and the index of agreement d2.
Figure 7 shows that T2m is often underestimated for Max Ψ and 
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 QH, and latent heat flux QE) for many cities and that because we here modify the furb 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 furb 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 furb 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 Δγ. Thus, higher-resolution categorical observations, such as the National Agricultural Imagery Program (NAIP), which has Δγ = 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 Curb. A more accurate representation of 
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 
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 
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 
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 |
| Curb | Urban land-cover class |
| furb | Urban fraction parameter |
 | Heterogeneous development density |
 | Heterogeneous development density for mosaic approach |
| h | Normalized Shannon entropy |
 | Conditional normalized Shannon entropy |
| Ns | Number of states (of p) |
| Nt | Number of tiles for mosaic approach |
| p() | Probability density function |
| Q* | Net radiation |
| QE | Latent heat flux |
| QH | Sensible heat flux |
| V | Generic model variable |
| Vtotal | Total aggregated V |
| Vurban | Urban contribution of V |
| Vnonurban | Nonurban contribution of V |
| w | Weighting coefficient |
| α | Normalized area fraction |
| Γ | Aggregated grid |
| Γ0 | Center of aggregated grid cell |
| γ | Fine grid |
| ΔΓ | Aggregated grid length scale |
| Δγ | Fine-grid length scale |
| μ | Mode of p(Ψ) |
 | Mode of p(Ψ | Curb) |
| Ψ | Percent developed imperviousness |
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