1

Downscaling near-surface wind over complex terrain using a physically-based statistical modeling approach

Hsin-Yuan Huang1, Scott B. Capps2, Shao-Ching Huang3, and Alex Hall2

1Joint Institute for Regional Earth System Science and Engineering,

University of California, Los Angeles 2Department of Atmospheric and Oceanic Sciences,

University of California, Los Angeles 3Institute for Digital Research and Education,

University of California, Los Angeles

Accepted by: Climate Dynamics

____________________ Corresponding author address: Hsin-Yuan Huang, 7343 Math Science Building, University of California, Los Angeles E-mail: hyhuang@ucla.edu

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Abstract 1

A physically-based statistical modeling approach to downscale coarse resolution 2

reanalysis near-surface winds over a region of complex terrain is developed and tested in 3

this study. Our approach is guided by physical variables and meteorological relationships 4

that are important for determining near-surface wind flow. Preliminary fine scale winds 5

are estimated by correcting the course-to-fine grid resolution mismatch in roughness 6

length. Guided by the physics shaping near-surface winds, we then formulate a 7

multivariable linear regression model which uses near-surface micrometeorological 8

variables and the preliminary estimates as predictors to calculate the final wind products. 9

The coarse-to-fine grid resolution ratio is approximately 10 to 1 for our study region of 10

southern California. A validated 3-km resolution dynamically-downscaled wind dataset is 11

used to train and validate our method. Winds from our statistical modeling approach 12

accurately reproduce the dynamically-downscaled near-surface wind field with wind 13

speed magnitude and wind direction errors of less than 1.5 ms-1 and 30 degrees, 14

respectively. This approach can greatly accelerate the production of near-surface wind 15

fields that are much more accurate than reanalysis data, while limiting the amount of 16

computational and time intensive dynamical downscaling. Future studies will evaluate 17

the ability of this approach to downscale other reanalysis data and climate model outputs 18

with varying coarse-to-fine grid resolutions and domains of interest. 19

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Keywords: Near-surface wind, dynamical downscaling, statistical downscaling, complex 21

terrain. 22

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1. Introduction 24

Wind flow patterns across the Earth’s surface are shaped by forces spanning a 25

vast range of scales throughout the atmosphere. Wind flow is a result of pressure 26

gradients associated with weather systems at the synoptic scale and near-surface thermal 27

contrasts due to horizontal changes in surface properties. While under the influence of 28

these synoptic scale forces, migrating air encounters finer-scale pressure gradients 29

resulting from topographic and surface roughness discontinuities. Accurate, high 30

resolution wind data over long enough time periods to compile climatological statistics is 31

important for climate change studies, pollutant dispersion evaluation, and wind energy 32

resource assessments. Near-surface wind speed is also critical to the operations of public 33

insurance (Changnon et al. 1999) and industrial utilities (Jungo et al. 2002). 34

Wind observations can be compiled to produce wind statistics. However, these 35

data are only valid at points where measurements are taken. The only way to obtain 36

complete spatial coverage is to use reanalysis data (e.g., North American Regional 37

Reanalysis, NARR; European Centre for Medium-range Weather Forecasts 40 Year 38

Reanalysis, ECMWF ERA-Interim). However, the resolution of reanalysis data ranges 39

from tens to hundreds of kilometers, resolving only major topographical features at best. 40

In reality, wind variations over a heterogeneous surface occur at much finer scales. A 41

downscaling technique deriving finer scale wind information from coarse scale data 42

would be advantageous if it were accurate because it would have much more complete 43

temporal and spatial coverage than reanalysis. 44

Typically there are two types of methods available to downscale meteorological 45

variables at resolutions finer than that of reanalysis data: dynamical and statistical 46

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downscaling. Both methods have been widely used in atmospheric and environmental 47

studies (Wilby and Wigley 1997). Using dynamical downscaling, one can obtain finer 48

scale results from a regional climate model (RCM, e.g., the Weather Research and 49

Forecasting model, WRF; the 5th-Generation Penn State/NCAR Mesoscale model, MM5, 50

etc.) forced by coarse resolution data as initial and boundary conditions. Depending on 51

the resolution of the regional model, this method can resolve complex topography and 52

heterogeneous surface conditions, providing more realistic finer scale wind information 53

for the domain of interest (Gustafson and Leung 2007). For example, Lebassi-Habtezion 54

et al. (2011) applied the Regional Atmospheric Modeling System to downscale low-level 55

winds and temperature for the Southern California region using the NCEP data as the 56

initial and boundary conditions. They found that mesoscale model results (e.g., near-57

surface wind and temperature) generally compared well to observations. 58

Statistical downscaling, on the other hand, derives statistical relationships 59

between local observations and coarse resolution reanalysis data using an empirical 60

approach or regression analysis (e.g., Gutierrez et al. 2004; Pryor et al. 2005). Some 61

recent studies also developed more complex approaches. For example, Sailor et al. (2000) 62

used neural networks to connect general circulation model data and surface wind 63

climatology observations, de Rooy and Kok (2004) applied a physically-based approach 64

to link turbulence similarity theory and near-surface wind, and Michelangeli et al. (2009) 65

developed a probability method to predict the temporal variability of wind distribution. 66

Nonlinear regression and multivariable linear regression methods have also been used in 67

some studies (e.g., Salameh et al. 2009; Curry et al. 2012; Haas and Pinto 2012). 68

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Dynamical and statistical downscaling methods each have their own advantages 69

and drawbacks. The dynamical downscaling technique provides detailed wind 70

information following fundamental physical principles. However, it is computationally 71

expensive and some parameterizations in regional atmospheric models have resolution 72

thresholds beyond which they are not designed to be used. For a given spatial resolution 73

and temporal period of interest, statistical downscaling methods require far fewer 74

computational resources. However, they require long duration historical data, which are 75

scarce. Also, it is not always clear that predictor variables giving the best fit for the 76

historical observations are appropriate for other time periods. Finally, spatial coverage in 77

statistical methods is limited to the spatial coverage of the data used to train the model. 78

Recent studies have started to merge the benefits of dynamical and statistical 79

downscaling methods (e.g., Vrac et al. 2007; Colette et al. 2012). In an ensemble 80

downscaling project using multiple RCMs, Yoon et al. (2012) compared precipitation and 81

temperature results from both dynamical and statistical downscaling methods for the cold 82

season over the United States. Their results suggest that a hybrid system integrating both 83

methods is able to increase the skill of model prediction. We introduce a technique which 84

combines benefits from both dynamical and statistical methods to downscale near-surface 85

winds across a study domain with complex terrain. The promise of the technique is that 86

dynamical downscaling provides more accurate and realistic winds than the driving 87

reanalysis data (e.g., Hughes and Hall 2010). We validate this promise further in this 88

work. However, dynamical downscaling is very computationally expensive. Our 89

approach is to perform only a limited amount of it, and then develop a physically-based 90

statistical approach that can mimic the dynamic model behavior. Then we can easily 91

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extend the dynamical outputs in time without spending significant computing time and 92

resources. Through the use of this statistical modeling technique, one can obtain fine-93

scale winds directly from reanalysis data. These are very similar to dynamically-94

downscaled results, but are at least an order of magnitude in computing time cheaper to 95

produce. 96

The structure of this manuscript is as follows. In Section 2, we introduce our 97

methodology, including the study domain, the dynamical downscaling simulation, and its 98

validation. Section 3 presents the statistical modeling approach which is comprised of a 99

physically-based multivariable linear regression. Section 4 evaluates the performance of 100

this statistical modeling compared to dynamically downscaled results, while section 5 101

summarizes the findings and outlines ongoing and future investigations. 102

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2. Methodology 104

2.1 Study domain and data 105

As shown in Figure 1, the study domain spans approximately 2.5° of latitude (San 106

Luis Obispo to San Diego) and approximately 6.5° of longitude (122 degree W over the 107

Pacific Ocean to just west of the Colorado River). Southern California is selected as the 108

domain of interest because of its complex topography and diverse surface types. Its 109

transverse and peninsular mountain ranges are geologically young and rugged, steering 110

and modulating wind flow throughout most of the region. Elevation across Southern 111

California ranges from zero to over 3000 m in the San Bernardino Mountains within a 112

distance of about 150 km. In between major complexes and the Southern California Bight 113

are vast urbanized basins separated by smaller mountains. Northeast of the rugged 114

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mountain ranges is the elevated Mojave Desert (~1200 m) with a relatively uniform 115

surface roughness and isolated, lower elevation mountain peaks. 116

In addition to the synoptic-scale pressure gradient and topographic steering 117

effects, near-surface wind speed magnitude is strongly affected by the underlying 118

momentum roughness length and thermal stability. Roughness length (as represented in 119

the dynamical model, discussed below in section 2.2) is proportional to obstacle height, 120

and ranges from 0.01 m over the ocean to 0.8 m in urban areas in this domain. The 121

mosaic of urban, agricultural and natural landscapes results in a diversity of surface 122

roughness and near-surface wind distributions. 123

Near-surface wind observations from the California Irrigation Management 124

Information System1 (CIMIS, black circles denote locations of stations in Figure 1) are 125

used to evaluate the performance of the dynamic downscaling simulation. Managed by 126

the California Department of Water Resources, CIMIS is a continuing program including 127

over 120 automated weather stations in the state of California since 1982. The primary 128

product of CIMIS is evapotranspiration, used to assist irrigators in efficient water 129

resource management. However, micrometeorological variables including wind speed at 130

2 m height (most importantly for this study) are fed into the evapotranspiration 131

calculation. Wind data from 25 CIMIS sites within the simulation domain are compared 132

against model output. 133

2.2 Dynamical downscaling simulations 134

The dynamical downscaling is performed using the National Center for 135

Atmospheric Research (NCAR) WRF Model Version 3.3 (Skamarock et al. 2008). We 136

use three nested domains. They have 58x51, 103x85 and 214x109 grid points at 27, 9 and 137 1 http://wwwcimis.water.ca.gov/cimis/

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3 km resolution, with the timestep of 90, 30, and 10 seconds, respectively. The outermost 138

domain (not shown) covers the entire state of California and a portion of the adjacent 139

Northeast Pacific Ocean, while the middle domain (also not shown) covers roughly the 140

southern half of the state. The innermost domain, with the finest grid resolution, is shown 141

in Figure 1. Only one-way nesting (from the outermost domain to the innermost domain) 142

is applied in the simulations. The vertical discretization has 44 levels up to an altitude of 143

50 mbar. Using the National Centers for Environmental Prediction’s 3-hourly, 32 km 144

resolution NARR2 data (Mesinger et al. 2006) as the initial and boundary conditions to 145

the outermost domain, we first perform two 1-year simulations (09/2009-08/2010 and 146

09/2010-08/2011) initialized at 00:00 UTC on August 30 for each year. The frequency of 147

model output is hourly. WRF requires 6-12 hours to fully spin up (Skamarock 2004; Lo 148

et al. 2008), thus data from the first two days are discarded as model spin up. 149

WRF provides multiple parameterization choices. Version 3.3 includes seven 150

shortwave and five longwave radiation schemes, 13 cloud microphysics models, nine 151

cumulus schemes, and 11 planetary boundary layer parameterizations. In this study we 152

use the Dudhia scheme (Dudhia, 1989) and the Rapid Radiative Transfer model (Mlawer 153

et al. 1997) for shortwave and longwave radiative flux calculations, respectively. While 154

the Purdue Lin scheme (Lin et al. 1983) is selected for cloud/liquid water microphysics 155

over the entire simulation domain, the Kain-Fritsch scheme (Kain 2004) is added to 156

include shallow cumulus in the two outer domains. The planetary boundary layer 157

parameterization is the Mellor-Yamada-Nakanishi-Niino (MYNN, Nakanishi and Niino 158

2004) scheme, based on a turbulent kinetic energy closure to estimate eddy diffusivity 159

and viscosity. Sea surface temperature is prescribed as the boundary condition over the 160 2 http://www.esrl.noaa.gov/psd/data/gridded/data.narr.html

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ocean. Over land, the NOAH land surface model is used with the 3-category urban 161

canopy model (Chen et al. 2011). A 3-D spatial analysis nudging technique (using the 162

NARR data) is applied on the outermost domain. Variables included in the nudging are 163

potential temperature, humidity, and wind components above the boundary layer top. 164

This analysis nudging restores the data in the outermost domain to the values of the 165

driving reanalysis data with a characteristic time scale. It not only constrains the error 166

growth in large-scale circulation during the simulation, but also improves the accuracy of 167

dynamic downscaling (Lo et al. 2008). This model setup has been used in a previous 168

study, Capps et al. (2014) to which the reader is referred to for more details. Output from 169

this dynamical downscaling calculation is used to provide a realistic distribution of near-170

surface wind for the development of the physically-based statistical downscaling 171

approach. 172

2.3 Evaluation of dynamical downscaling results 173

To verify that dynamic downscaling provides more realistic winds than reanalysis 174

data, we first compare NARR and WRF daily mean wind speeds against CIMIS 175

observations at the grid points closest to the CIMIS station locations. Because the 176

observations are collected at 2 m above the land surface, NARR and WRF 10 m wind 177

speeds are extrapolated to 2 m using the log-law. As seen in Figure 2a, an acceptable 178

agreement exists between NARR and observed winds. The bulk of the temporal 179

correlation coefficients are in the 0.6-0.7 range, indicating that the large-scale NARR 180

wind field is able to explain as much as 50% of the variance in CIMIS observations. In 181

comparison, WRF winds match observed CIMIS winds more closely with respect to 182

NARR. Both the spatial and temporal variations are much better correlated (Figure 2b). 183

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In the case of WRF, average values of daily mean wind speed correlation coefficient, 184

root-mean-square-error and bias across 25 sites are 0.80, 0.56 (ms-1) and 0.05 (ms-1), 185

respectively. 186

Time series of wind speed at two selected sites (#62 in Orange County and # 134 187

in the Mojave Desert, shown in Figure 1) are shown in Figures 2c and 2d, where red 188

spots and blue lines are observations and WRF output, respectively. Both plots show a 189

good temporal agreement between simulation outputs and observations. The correlation 190

coefficient and root-mean-square-error at site #62 (#134) are 0.78 (0.77) and 0.62 (1.08) 191

ms-1, respectively. More validation of this dynamical simulation configuration using other 192

observations (e.g., data obtained from the National Climatic Data Center) can be found in 193

Capps et al. (2014). Results of this validation give us confidence that the spatial and 194

temporal wind variations in WRF are reasonably realistic, and more importantly that 195

WRF downscaling provides a more realistic wind field compared to NARR. Therefore, it 196

is worthwhile to build a physically-based statistical modeling framework to reproduce the 197

WRF output. 198

199

3. Physically-based statistical modeling approach 200

In this section, we describe the physically-based statistical modeling approach 201

used to downscale daily mean near-surface wind from 32-km resolution NARR data 202

(hereafter referred to as “coarse grid”) to the 3-km resolution used in WRF simulation 203

(hereafter referred to as “fine grid”). The process involves two steps: First, we generate 204

preliminary estimates using Monin-Obukhov similarity theory (MOST, Monin and 205

Obukhov 1954). Second, the preliminary estimates are used in conjunction with other 206

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relevant surface and micrometeorological variables (e.g., sea-level pressure and surface 207

fluxes) of NARR data in a multivariable linear regression model to achieve final near-208

surface wind estimates. 209

3.1 Preliminary estimate 210

Heterogeneities in surface characteristics (e.g., topography, roughness, vegetation 211

type, etc.) play an important role in shaping near-surface meteorology, including near-212

surface u- and v-wind components, humidity and temperature. For example, using a 213

series of large-eddy simulation experiments to investigate a realistic convective boundary 214

layer, Huang and Margulis (2009) found that surface heterogeneity significantly impacts 215

both thermal and momentum blending heights. Momentum blending height is a vertical 216

length scale above which the influence of surface characteristics on momentum terms 217

(e.g., horizontal velocity) vanishes below some specific value (Wieringa 1986). 218

Following the concept introduced in de Rooy and Kok (2004), which reduced errors in 10 219

m wind estimates downscaled from a coarse resolution model, we assume that the 220

variation of near-surface wind below the blending height follows Monin-Obukhov 221

similarity theory. The wind magnitude ( hu ) at height hz above the surface can be written 222

as: 223

*

0 mo

lnL

h hh M

z zuu

k z

, (1) 224

where *u is friction velocity, k is the von Karman constant, 0z is surface momentum 225

roughness height, and M is the stability function which is a function of Obukhov length 226

moL (Garratt 1994). 227

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We can use Eqn. (1) to formulate wind speeds at 10 m (i.e., 10u at 10z ) and at the 228

blending height (i.e., bhu at bhz ) in the coarse grid model. We then rearrange the two 229

equations as: 230

0 mo

10

0 mo

ln 10 10 L

ln L

L LML L

bh L LM

zu u

bh z bh

, (2) 231

where the superscript L represents data with coarse grid model. One can also write the 232

same equation for data from the fine grid model with superscript S . Applying some 233

algebraic operations on these two equations for coarse and fine grid models, we can 234

rewrite 10 m wind at the fine grid resolution as: 235

10

0 mo 0 mo

10

0 mo 0 mo

Preliminary estimate

ln 10 10 L ln 10 10 L

ln L ln L

S

S S L LM ML S L

bh bhS S L LM M

u

z zu u u

bh z bh bh z bh

. (3) 236

Note that 10Su is also our “preliminary estimate”. 237

To simplify Eqn. (3), we invoke two assumptions. The first is that the blending 238

height is fixed and is similar in the coarse and fine grid models. This means the wind 239

magnitudes at the blending height are similar in both coarse and fine grid models (i.e., 240

S Lbh bhu u ) and that the wind flow above the blending height is not significantly affected 241

by the surface characteristics. Instead, it is dominated by atmospheric flow at larger-242

scales. Using a similar concept to downscale near-surface wind, various values of 243

blending height have been used (e.g., de Rooy and Kok (2004) used 140 m, Strassberg et 244

al. (2008) used 65 m). However, McNaughton and Jarvis (1984), who used 100 m, 245

suggest that the exact value of selected blending height is not very critical because the 246

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changes in the vertical gradient are small around these heights. For this study, we select a 247

blending height of 100 m. Thus, Eqn. (3) can be rewritten as: 248

10

0 mo 0 mo

10 100

0 mo 0 mo

Preliminary estimate

ln 10 10 L ln 10 10 L

ln 100 100 L ln 100 100 L

S

S S L LM ML L

S S L LM M

u

z zu u

z z

. (4) 249

The second assumption is that the stability function is negligible. This assumption 250

may be reasonably accurate because, if we use the daily mean wind, the instability effect 251

on the wind profile during daytime may be roughly offset by the stability effect at night. 252

Additionally, the order of the stability function is close to zero for a nearly neutral 253

condition (Garratt 1994). 254

With these two assumptions, Eqn. (4) can be further simplified as: 255

0 0

10 10 100

0 0

0 0

10 100

0 0

ln 10 ln 10Preliminary estimate

ln 100 ln 100

ln10 ln ln .

ln 100 ln 100

S L

S L L

S L

L S

L L

S L

z zu u u

z z

z zu u

z z

(5) 256

On the right-hand-side of Eqn. (5), values of surface roughness for the coarse grid model 257

( 0Lz from NARR) and fine grid model ( 0

Sz from WRF/local observations) are known and 258

prescribed, and 10Lu (NARR 10 m wind) and 100

Lu (NARR 100 m wind) are obtained from 259

NARR wind speed data at 100 m height using a cubic spline interpolation. Eqn. (5) is 260

essentially an expression to recover fine resolution mean wind velocity using fine 261

resolution roughness information and coarse resolution velocities at different heights 262

based on MOST. This preliminary estimate from Eqn. (5) is a valuable first step before 263

further statistical modeling. It incorporates fine-scale variations in the most important 264

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fixed physical parameter affecting 10 m wind ( 0z ), and transforms wind speed 265

information from the blending height to 10 m. Note that the value in curly braces in the 266

second right-hand-side term of Eqn. (5) can be negative when 0Lz is less than 0

Sz , which 267

is usually seen in coastal and urban areas in our study domain. However, this condition 268

does not result in a negative preliminary estimate as long as the ratio of 100 10L Lu u does not 269

exceed 7.6. We also perform additional examinations (not shown) to consider all possible 270

combinations of an moL ranging from -200 to 200 m (except an interval between -10 and 271

10 m representing a neutral condition) and a 0z ranging from 0.05 to 0.425 m for the 272

similarity theory, and the result shows that the value of 100 10L Lu u is less than this critical 273

value for all cases. 274

3.2 Example of preliminary estimate 275

Figures 3a and 3b present an example of the standard NARR 10 m wind 276

magnitude on October 15, 2009, together with the NARR surface roughness. This shows 277

NARR’s poor representation of the coastline due to its coarse resolution (~32 km). The 278

NARR surface roughness map is also unable to accurately represent the highly 279

heterogeneous land use categories in the study domain. Urban and mountain zones, where 280

high roughness length should be observed due to significant buildings and tall forests, are 281

not clear. The 3-km resolution of surface roughness used in the WRF simulation is 282

illustrated in Figure 3c. In this case, the coastline is clearly more realistic. Over land, 283

urban areas have the highest roughness length (0.8 m) while the Mojave Desert has the 284

lowest (less than 0.1 m). In this study, using the default setting in WRF model, we simply 285

assign one roughness length for each land use category and one for the ocean surface (In 286

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reality, roughness lengths over land may also depend somewhat on seasonally-varying 287

vegetation height and those over the ocean surface may depend somewhat on wave 288

height). 289

The preliminary estimate of 10 m wind for October 9, 2009 using Eqn. 5 is shown 290

in Figure 3d. The wind speed magnitude patterns in Figure 3d show a correspondence to 291

those in Figure 3a, but with a discernable modulation of the coarser winds by the 292

underlying higher resolution land category pattern. A clearer portrayal of this modulation 293

is seen in Figure 3e, which shows the difference between the preliminary estimate and 294

NARR 10 m wind interpolated to WRF grid resolution. Smaller values of the preliminary 295

estimate values are simulated in urban and high elevation areas, while larger values are 296

seen in the high desert. In major metropolitan areas, the wind speed reduction can be 297

larger than 60 % (Figure 3f). Furthermore, the difference in roughness length between the 298

land and sea creates a discernible gradient of wind magnitude across the coastline. Next, 299

we will incorporate these preliminary estimates into a multivariable linear regression 300

model to further obtain the final statistical downscaled 10 m wind field. 301

3.3 Multivariable linear regression model 302

The second step of this approach is the development of a multivariable linear 303

regression using the preliminary estimate and NARR near-surface meteorological 304

variables as inputs. The meteorological variables are selected to include influences on 305

winds that were missed in the preliminary estimate. 306

Since the impact of thermal stability on 10 m wind is neglected in the previous 307

step (i.e., neutral condition as assumed), the first variable to include in the regression 308

model is the thermal flux. Since latent and sensible heat fluxes both contribute to the 309

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energy fluxes from the surface to the atmospheric boundary layer, we combine them to 310

create our first variable. This surface buoyancy flux is defined as: 311

0.61 ap v

H LEw T

c l

, (6) 312

where H is sensible heat flux, LE is latent heat flux, is near-surface air density, aT is 313

air temperature, and pc and vl are the specific heat of air and the latent heat of 314

vaporization of water, respectively. Based on the momentum equation of fluid mechanics, 315

the second variable we select in this regression model is spatial difference of sea level 316

pressure ( seaP ) which plays an important role in both wind speed and direction. As one 317

of many standard NARR outputs, pressure at surface is reduced to sea level using the 318

Mesinger method (Mesinger and Treadon 1995). For each WRF grid point, mean sea 319

level pressures of the four closest surrounding NARR points are interpolated to estimate 320

its pressure using an inverse distance weighting method. Then, we define the sea level 321

pressure difference between the NARR point with the highest pressure (among these four 322

surrounding NARR points) and the WRF point as seaP , which represents the change of 323

sea level pressure resulting in a change in wind velocity. 324

To illustrate the relationship between these two meteorological variables and the 325

variable we are interested in predicting (i.e., WRF 10 m wind speed), figure 4 shows 326

maps of temporal correlation coefficient between the WRF dynamically downscaled 10 327

m wind speed and NARR w and seaP . Due to the difference in grid resolutions 328

between WRF and NARR data, a simple 2-D linear interpolation is applied to NARR 329

w and seaP prior to calculating these correlations. High positive correlations between 330

10 m wind speed and surface buoyancy flux are seen over ocean and desert areas. 331

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Negative correlations between surface wind and buoyancy flux are seen over land, 332

especially near major passes (Figure 1). This is primarily due to the occurrence of fast 333

offshore winds through these passes during autumn and winter months, when buoyancy 334

flux is relatively small. High correlations are generally seen in most locations in the 335

correlation map of sea level pressure gradient (Figure 4b). Thus, in addition to the 336

preliminary estimate, these two variables are selected in our regression model. 337

Directly using variables with different units in a regression system could result in 338

a badly-conditioned coefficient matrix. This may be especially a problem in our case 339

because the magnitudes of the pressure difference or buoyancy flux are not of the same 340

order and do not have the same unit as wind speed. So, borrowing from the idea of the 341

Buckingham π dimension analysis, we apply a dimensional analysis to convert the units 342

of all meteorological variables to be consistent with that of wind speed (ms-1). We can 343

derive a new variable associated with the pressure difference 1 2

1 seaP , where 1 344

is a variable with units of ms-1, and is air density (kgm-3) which depends on location. 345

We also can apply the same approach to derive another variable corresponding to the 346

buoyancy flux 2 0w , where 0 is a reference temperature of 290 K (close to the 347

annual averaged temperature in our study domain). Using these proxy variables ensures 348

the stability of the regression matrix. 349

Therefore, we use these three variables (i.e., 1 , 2 and preliminary estimate) to 350

construct a multivariable linear regression model to produce a simple least-square 351

estimate of WRF wind speed. A four-quadrant inverse tangent function is applied to 352

individual wind components (i.e., u and v ) of both NARR data and WRF dynamic 353

results to calculate wind directions between -180 and 180 degrees. An additional simple 354

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linear regression model is trained and used to relate the wind directions between WRF 355

output and NARR data. 356

357

4. Results of statistical modeling 358

Wind estimates using the physically-based statistical modeling are presented and 359

compared against actual WRF dynamical downscaling outputs in this section. Two 360

experiments are performed. In the first experiment, we use the first year (09/2009-361

08/2010) of dynamically-downscaled data to develop and train the statistical model, and 362

then we evaluate the model performance over the second year (09/2010-08/2011). Then 363

the training and evaluation periods are swapped in the second experiment. The second 364

year is used as the training period and the first year is as the evaluation period. Such 365

swapping of experiments allows us to examine the statistical relationships between 366

predictors and estimates. In the following paragraphs we first show an example of 367

statistically modeled wind estimates and further compare against dynamical downscaling 368

results in detail. 369

4.1 Example of wind estimate 370

We first provide an example of dynamically downscaled and statistically modeled 371

wind speed and direction distributions selected from a day in November 2010 to give a 372

flavor of the results (Figure 5). The statistical estimates are based on the training period 373

of 09/2009-08/2010. Overall, the statistically modeled results (Figure 5b) resemble the 10 374

m wind pattern of the dynamical downscaling simulation (Figure 5a) closely. The wind 375

field is that of a typical offshore event where katabatic winds flow off the high desert, 376

descend as they cross the mountains and funnel through the passes. Particularly fast 377

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offshore winds are blowing from the Newhall Pass out over the Santa Clarita Valley, the 378

Oxnard Plain, and the adjacent ocean. Over the open ocean the wind is generally 379

northerly, with a decrease in magnitude near the coast and islands. Significant wind 380

direction changes are seen in the channels between islands and the mainland. Over land, 381

slower wind speeds are seen in most areas, while faster winds occur in major passes and 382

west of the San Bernardino and Santa Ana Mountains. Slower winds are also seen in 383

industrial and residential areas due to the relatively high surface roughness associated 384

with these land use categories. 385

Consistent with the geographical patterns of wind speed, there are remarkable 386

similarities in the wind distributions between dynamically downscaled (Figure 5c) and 387

statistically modeled (Figure 5d) cases. The statistically modeled wind field slightly 388

overestimates the frequency of slower winds, especially those with medium magnitude 389

(4-8 ms-1). In general, as shown in the spatial map and wind rose distribution, the 390

statistical modeling approach reproduces the dynamically downscaled wind results well. 391

4.2 Map of temporal error statistics 392

Maps of error statistics verifying the overall skill of the statistical approach are 393

shown in Figure 6. Two swapped experiments are performed as mentioned previously 394

(i.e., while one year is treated for training, the other is used for evaluation), and the 395

following results are averages of the two evaluation periods. Panel a) of this figure shows 396

the correlation coefficient between the dynamical output and the statistical prediction. In 397

general, large regions of high wind speed correlations (around 0.9) are seen over the 398

Mojave Desert and the ocean surface. Somewhat slightly lower correlations in the 0.7-0.8 399

range are seen in urban areas and coastal valleys. The Mojave Desert and ocean consist of 400

20

relatively homogeneous surface characteristics with small sub-grid scale variability. The 401

NARR winds themselves are likely better correlated with WRF wind at these locations, 402

and therefore it is expected that the statistical modeled winds would be better correlated 403

to the dynamical results over these regions. On the other hand, winds over coastal, 404

mountainous and urban regions could be significantly affected by small-scale variability 405

of surface characteristics. Thus, the more difficult it is for statistical model to accurately 406

predict winds over such complex regions. This result is consistent with the correlation 407

coefficients of predictors shown in Figure 4. 408

Compared to dynamical downscaling, the absolute error of the statistical modeled 409

winds is larger over ocean than land (Figure 6b). The reason for this is that wind speed 410

increases with a decrease in surface roughness length (see the map in Figure 3c), creating 411

larger wind speeds over ocean than land. Since errors in statistical estimates ought to be 412

roughly proportional to wind magnitude, larger errors are also seen over the ocean. 413

Larger errors are seen over the deserts as well, as wind speeds are also faster there. 414

Finally, slightly larger errors are seen in areas near mountains, which are poorly 415

represented in coarse resolution data. The relative error map shown in Figure 6c also 416

confirms this. While the ocean and deserts have smaller relative errors, slightly larger 417

relative errors are seen over land between the coast and mountains. 418

Overall, with a similar flow pattern as that shown in the WRF simulation and 419

acceptable differences in both wind speed and direction, the statistical model reproduces 420

the spatial distribution of near-surface wind over both ocean and land surfaces well. 421

4.3 Time series of spatial error statistics 422

21

Time series of statistics documenting the spatial relationship between dynamically 423

downscaled and statistical modeled winds are plotted in Figure 7. The thin lines in light 424

color and the thick lines with markers represent statistics based on daily and monthly 425

averages, respectively. Consistent with the example shown in Figure 5, very high 426

correlations (red line) are seen in wind speed (Figure 7a). Most values of the daily 427

correlation coefficient are between 0.70 and 0.95, while all monthly mean values are 428

higher than 0.8. The absolute value of the wind speed difference is shown as the blue 429

line, which represents the mean of the absolute value of each difference between the 430

dynamically downscaled winds and the statistically modeled wind estimates. The result 431

shows that the largest error in monthly mean wind estimate is less than 2 ms-1, while the 432

overall average is about 1.2 ms-1. The green line shows the bias in wind speed estimates. 433

Here we define the bias as the simple arithmetic mean of difference, also referred to as 434

the mean signed difference. The bias varies within a range between -1.5 and 1.5 ms-1, 435

with monthly means close to zero, illustrating the accuracy of this physically-based 436

statistical approach. 437

Similar results are seen in the comparison of wind direction (Figure 7b). High 438

correlations with values hovering around 0.9 are seen except for a few days with values 439

lower than 0.7. The absolute difference of wind direction (blue line) is typically 20°, with 440

the largest error being no larger than 40°. In general, the wind direction estimate is more 441

accurate in summer than winter. A possible reason could be that winds blow consistently 442

southward (i.e., northerly winds) during summer in our study domain. In winter, winds 443

blow offshore during Santa Ana events and onshore during precipitation events, 444

occasionally disturbing the normal wind regime (Conil and Hall 2006). Because the wind 445

22

direction anomalies are larger in winter, the error of the statistical model may also be 446

larger. A similar story is seen in plot of bias (green line). The range of the daily mean 447

bias is 30°, and the bias in summer is significantly smaller than in winter. 448

4.4 Error statistics in terms of surface property 449

Figure 8 shows the error statistics binned by surface elevation. The absolute value 450

of wind speed error (Figure 8a) shows that, excluding data over the ocean surface (i.e., 451

elevation<5 m), the error increases systematically with elevation. Relative error (Figure 452

8b) is generally less than 15%, and is insensitive to elevation change, consistent with 453

Figure 6c (i.e., the errors are roughly proportional to wind magnitude). Errors in the wind 454

direction estimate slightly increase with surface elevation (Figure 8c), probably due to 455

dynamical effects associated with complex terrain. In mountain regions compressed 456

winds are found on the windward side of the mountains, and the flow then expands 457

downstream while flowing over the lee side of the mountains. Since these effects are 458

unrelated to surface roughness, surface buoyancy, or surface pressure, the statistical 459

model may have difficulty capturing them. 460

Finally, we compare the statistical modeling estimates against both the observed 461

CIMIS data and WRF dynamical downscaled results. Generally, the inter-daily wind 462

speed variability estimated by the statistical model is higher compared to WRF (Figures 463

9a (site #62) and 9b (site #134)). The statistical model also frequently underestimates 464

wind speeds at sites #134 and #64 when compared to both WRF and the observations. 465

The root-mean-square-error between observations and statistical estimates is 0.83 and 466

1.39 ms-1 at site #62 and #134, respectively. These root-mean-square-errors are about 0.2-467

0.3 ms-1 higher compared to WRF (Section 2.3). Similar results are also seen at additional 468

23

sites (site #64 in Figure 9c and site #208 in Figure 9d). It is not surprising that the 469

statistical model performs slightly worse than the WRF dynamically-downscaled winds. 470

However, because the statistical approach is trained on the WRF output, the quality of 471

statistical estimates is comparable to the WRF results. This result implies the dependence 472

of the statistical approach on the dynamical downscaling results. If the dynamic 473

downscaling technique (i.e., WRF model) is not able to closely reproduce the 474

observations, an even larger error and bias will occur in the statistical estimates. 475

4.5 Contribution of regression variables 476

The bar plot in Figure 10 shows the average contribution from each regression 477

variable to the statistical wind speed estimate in terms of land surface cover category. 478

Data are predictor variables multiplied by their own regression coefficients. It is clear 479

that, for most land cover types, the contribution of the preliminary estimate (red bars) is 480

significantly larger, while the contribution of 2 (corresponding to the buoyancy flux, 481

green bars) is the smallest. This result is expected because the fine grid surface roughness 482

length that participates in the preliminary estimate procedure through similarity theory 483

plays the most important role in determining 10 m wind speed. Meanwhile, this study 484

estimates the daily mean wind, where the daytime stability effect may be offset by night 485

time. The overall proportions of contribution from the preliminary estimate, the variable 486

associated with the pressure difference ( 1 ), and the variable associated with the 487

buoyancy flux ( 2 ) are roughly about 65%, 25%, and 10%, respectively. It is possible 488

that the buoyancy flux (i.e., thermal stability condition) would be more significant if one 489

estimated the sub-daily wind field using this statistical approach, rather than the daily-490

mean seen in Figure 10. Furthermore, the consideration of pressure difference could be 491

24

essential for places with complex terrain or land use change. An example is areas with 492

mixed land types, for instance, where residential regions (with high roughness) are 493

intermixed with hills and high mountains in our study domain. 494

495

5. Conclusions 496

We develop and apply a physically-based statistical modeling approach to 497

downscale near-surface wind from NARR data over the complex terrain of Southern 498

California. This approach is comprised of two principal steps. First, we apply Monin-499

Obukhov similarity theory to generate preliminary estimates. These preliminary estimates 500

substantially correct wind speed over areas where there is a significant mismatch in 501

roughness length between the coarse and fine resolution data. Then, to obtain the final 502

wind estimate, we construct a multivariable linear regression including the preliminary 503

estimate and two meteorological variables that have significant impacts on near-surface 504

wind speed and direction. In addition to mimicking the momentum equation of wind 505

physics, dimensional analysis of micrometeorological variables in the multivariable linear 506

regression approach provides unit consistency. 507

Our statistical estimates accurately reproduce the 10 m wind fields simulated in 508

the dynamic downscaling. The absolute value of daily averaged wind speed estimates is 509

smaller than 1.5 ms-1, and most errors in wind direction are less than 30 degrees across 510

the entire simulation period. The accuracy of the wind estimate using this physically-511

based statistical modeling approach tends to degrade somewhat over highly complex 512

terrain. Analysis of regression variables also shows that, for the daily-mean wind estimate 513

25

in this study, the contribution of the preliminary estimate dominates the magnitude of 514

statistical wind speed and the correction of the buoyancy flux is less important. 515

In addition to near-surface wind, this physically-based statistical modeling 516

approach could be applied to other variables as long as there is a physical relationship 517

between the variable of interest and other micrometeorological characteristics. In addition 518

to wind resource applications, fine scale wind products provided here can also be used to 519

improve estimates of related meteorological variables (e.g., surface fluxes). In this study, 520

the resolution ratio between reanalysis data and dynamic downscaling in the statistical 521

model is about a factor of ten (i.e., 32-km NARR data is downscaled to 3-km). However, 522

in many climate studies and operational applications, coarse resolution data could be 523

General Circulation Model output, where the grid resolution could be on the order of 100 524

km. Such coarse resolution data resolves a limited amount of physical processes, with 525

coarser spatial and temporal resolutions, possibly reducing the amount of information 526

available for the relationships in this physically-based statistical modeling approach. This 527

may limit the performance of the statistical approach. Thus, in subsequent work, impacts 528

of resolution difference between coarse and fine data on this statistical approach will be 529

examined. To extend the application of proposed approach, we are using this framework 530

to estimate near-surface winds for a 20-year period for both current and future climates. 531

We would also like to compare our results with those using other state-of-the-art 532

statistical downscaling approaches in a follow-up work. 533

534

Acknowledgments. 535

26

This work was supported by the Department of Energy Grant #DE-SC0001467 and the 536

California Institute for Energy & Environment Grant #POEA01-A02. The authors would 537

like to thank the reviewers for their helpful comments.538

27

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631

Figure 1. Classifications of land cover for the study domain, which is the innermost, 3-632

km resolution domain of our WRF simulation. Black lines represent elevation contour 633

lines of 1000 m (thin) and 2000m (thick) above sea level. Black markers are locations of 634

the CIMIS observation sites. Number 62 and 134 are two CIMIS sites shown in Figure 2, 635

and 64 and 208 are two additional sites shown in Figure 9. The percentages next to each 636

land category on the right indicate the fraction of the entire simulation domain 637

corresponding to the land category. A water surface (ocean and lake, shown in white) 638

occupies about 36% of the study domain. Gray lines indicate the borders of Los Angeles 639

and Orange counties.640

31

641

Figure 2. 2-m daily mean wind measurements versus the model predictions: scatter plots 642

of daily mean CIMIS observed and closest a) NARR and b) WRF grid cell wind speed 643

over the simulation period. Marker locations designate the means over the two-year 644

period and their colors represent the correlation coefficients of daily variability (as shown 645

in the colorbar below). Time series of CIMIS observations (red circles) and WRF outputs 646

(cyan lines) selected from c) site 62 and d) site 134 (site locations shown in Figure 1). 647

32

648

Figure 3. a) Daily-averaged NARR 10 m wind (ms-1) for October 15, 2009, b) NARR 649

surface roughness (m), c) WRF surface roughness (m), d) the preliminary estimate of 10 650

m winds (ms-1) for October 15, 2009, e) wind magnitude difference (ms-1) and f) 651

percentage difference (%) between the preliminary estimate and NARR data, interpolated 652

to WRF grids.653

33

654

Figure 4. Maps of the correlation coefficient between daily-averaged WRF 10 m wind 655

speed and NARR daily averaged a) surface buoyancy flux, and b) spatial difference of 656

sea level pressure.657

34

658

Figure 5. Example of dynamically downscaled and statistically modeled winds for a day 659

in November 2010: a) dynamically downscaled wind speed (ms-1), b) statistically 660

modeled wind speed (ms-1), c) dynamically downscaled wind rose plot, and d) 661

statistically modeled wind rose plot. In panels a) and b), arrows illustrating wind speed 662

and direction are shown for every 10 WRF model grid points for clarity. The wind rose 663

plot uses the meteorological convention.664

35

665

Figure 6. Maps of a) correlation coefficient, b) absolute error (ms-1), and c) relative error 666

(%) between dynamically downscaled and statistically modeled daily-averaged wind 667

speed. Data are averages of the two swapped training and testing experiments. See text 668

for details.669

36

670

Figure 7. Time series of error statistics in statistically modeled results: a) wind speed 671

(ms-1) and b) wind direction (in degrees) estimates. Each data point represents a 672

comparison of spatial variations in wind at any given time. Red lines plot the correlation 673

coefficients, blue lines, the mean absolute difference and green lines, the bias. Thin lines 674

and thick lines with markers represent daily and monthly averages, respectively. Data are 675

averages of the two swapped training and testing experiments.676

37

677

Figure 8. Statistics of a) absolute value of wind speed difference (ms-1), b) relative error 678

(%), and c) absolute value of wind direction difference (º) binned by surface elevation. 679

Red lines and blue boxes represent the median and central 50% of data, respectively, and 680

the whisker length represents a range of approximately two standard deviations. 681

38

682

Figure 9. Time series of 2-m daily mean wind CIMIS observations (red circles), WRF 683

winds (cyan lines), and statistical modeling estimates (blue lines) selected from a) site 62, 684

b) site 134, c) site 64, and d) site 208. Site locations are shown in Figure 1. 685

39

686

Figure 10. Averaged proportion (%) of contribution from each regression variable (i.e., 687

1 , 2 and preliminary estimate) to statistical modeling wind speed estimate. 688